Galaxy bimodality due to cold flows and shock heating
Avishai Dekelstar and Yuval Birnboimstar
ABSTRACT
We address the origin of the robust bimodality observed in galaxy properties about a characteristic
 stellar mass ~3 x 1010 Mcircledot. Less massive galaxies tend to be ungrouped blue star
forming discs, while more massive galaxies are typically grouped red old-star spheroids.
Colour-magnitude data show a gap between the red and blue sequences, extremely red luminous
 galaxies already at z ~ 1, a truncation of today's blue sequence above L*, and massive
starbursts at z ~ 2-4. We propose that these features are driven by the thermal properties of the
inflowing gas and their interplay with the clustering and feedback processes, all functions of
the dark matter halo mass and associated with a similar characteristic scale. In haloes below a
critical shock-heating mass M shock lessorsimilar 1012 Mcircledot, discs are built by cold streams, not heated by a
virial shock, yielding efficient early star formation. It is regulated by supernova feedback into
a long sequence of bursts in blue galaxies constrained to a 'fundamental line'. Cold streams
penetrating through hot media in M greaterorequalslant M shock haloes preferentially at z greaterorequalslant 2 lead to massive
starbursts in L > L* galaxies. At z < 2, in M > M shock haloes hosting groups, the gas is heated
by a virial shock, and being dilute it becomes vulnerable to feedback from energetic sources
such as active galactic nuclei. This shuts off gas supply and prevents further star formation,
leading by passive evolution to 'red-and-dead' massive spheroids starting at z ~ 1. A minimum
in feedback efficiency near M shock explains the observed minimum in M/L and the qualitative
features of the star formation history. The cold flows provide a hint for solving the angular
momentum problem. When these processes are incorporated in simulations they recover the
main bimodality features and solve other open puzzles.
Key words: shock waves - cooling flows - galaxies: evolution - galaxies: formation - galaxies:
haloes - dark matter.
1 INTRODUCTION
Observations reveal a robust bimodality in the galaxy population,
being divided into two classes, the 'blue' and 'red' sequences, at
a characteristic stellar mass M s,crit similarequal 3 x 1010 Mcircledot. This corresponds
 to a dark halo mass M crit lessorsimilar 1012 Mcircledot and a virial velocity
V crit similarequal 120 km s-1 today. Less massive galaxies tend to be blue,
star forming discs residing in the 'field'. Their properties are correlated
 along a 'fundamental line' of decreasing surface brightness,
internal velocity and metallicity with decreasing luminosity. Galaxies
 above M crit are dominated by spheroids of red, old stars, with
high surface brightness and metallicity independent of luminosity.
They tend to reside in the high-density environments of groups and
clusters and they preferentially host active galactic nucleus/nuclei
(AGN/AGNs).
Current models of galaxy formation have difficulties in reproducing
 this bimodality and the broad colour distribution observed.
In particular, the extremely red bright ellipticals which start showing
 up already at z ~ 1 are not predicted. They require efficient
star formation at earlier epochs, followed by an effective shutdown
of star formation in massive galaxies. The observations also reveal
very blue galaxies in excess of the predictions, indicating repeating
starbursts over the lifetimes of galaxies. Today's blue sequence is
non-trivially truncated at the bright end, while at z greaterorequalslant 2 there are
indications for very luminous starbursts in big objects, both posing
severe theoretical challenges.
1.1 The observed bimodality
The bimodality or transition in galaxy properties is observed in many
different ways. We list here the main relevant observed features1
which we address in this paper (Section 6).
(i) Luminosity functions. Blue galaxies dominate the stellar
mass function below M s,crit while red galaxies take over above it

[Bell et al. 2003c; Baldry et al. 2004, in Sloan Digital Sky Survey
(SDSS) and two-Micron All Sky Survey (2MASS)]. The transition
occurs slightly below L*, the characteristic luminosity of the brightest
 disc galaxies, beyond which the luminosity function drops.
(ii) Colour-magnitude. A colour bimodality shows robustly in
Colour-magnitude diagrams, where the galaxies are divided into
a blue sequence and a red sequence separated by a gap. In SDSS
(Blanton et al. 2005; Baldry et al. 2004), the gap is at u - r ~ 2. The
colour distribution is non-trivially broad, with the red tip stretching
beyond u - r = 2.5 and the blue tail reaching well below u - r = 1.0.
(iii) Star formation rate. The current star formation rate (SFR),
and the typical age of the stellar population, show a robust bimodality
 about M s,crit. The less massive galaxies are dominated by young
populations, while the more massive galaxies are dominated by old
stars [Kauffmann et al. 2003b; Madgwick et al. 2003b, in SDSS and
two-degree Field (2dF)], in agreement with the colour bimodality. A
similar bimodality is seen in the gas-to-stellar mass fraction, which
is high in the blue sequence and low in the red sequence, steeply
increasing with stellar mass below M s,crit, and only moderately so
above it (Kannappan 2004, in SDSS+2MASS).
(iv) Colour-magnitude at z ~ 1. The colour bimodality is similar
 back to z ~ 1.5 (Bell et al. 2004, in COMBO17). Extremely red
massive galaxies exist at the bright tip of the red sequence already
at z ~ 1 (e.g. Moustakas et al. 2004). Very blue small galaxies indicating
 starbursts show in the blue sequence (e.g. Ferguson & Babul
1998; Fioc & Rocca-Volmerange 1999).
(v) Massive starbursts at high z. Very luminous and massive
dusty objects are detected at z ~ 2-4, indicating an excessive activity
of star formation in surprisingly big objects (Smail et al. 2002;
Chapman et al. 2003, 2004; Shapley et al. 2004, LBG and SCUBA
sources).
(vi) Star formation history. The cosmological history of SFR
has a broad maximum near z ~ 1-2, followed by a sharp drop from
z ~ 1toz = 0 (e.g. Madau et al. 1996; Dickinson et al. 2003;
Giavalisco et al. 2004; Hartwick 2004; Heavens et al. 2004). Still,
about half the stars in today's spirals seem to have formed after z ~
1, e.g. in luminous infrared galaxies (LIRGs) near M s,crit (Hammer
et al. 2005). Massive galaxies tend to form their stars earlier than
smaller galaxies ('downsizing' Thomas et al. 2005).
(vii) Bulge-to-disc ratio. The galaxy bulge-to-disc ratio shows
a transition from disc dominance in the blue sequence below M s,crit
to spheroid dominance in the red sequence (Blanton et al. 2005;
Kauffmann et al. 2003b, in SDSS).
(viii) Environment dependence. The distributions in colour and
SFR depend strongly on the galaxy density in the ~1 Mpc vicinity:
the blue and red sequence galaxies tend to populate low- and highdensity
 environments, respectively (Blanton et al. 2005; Hogg et al.
2003; Balogh et al. 2004; Blanton et al. 2004; Kauffman et al. 2004,
in SDSS). The colour-environment correlation is stronger than the
morphology-density relation Dressler (1980).
(ix) Halo mass and HOD. The environment density is correlated
 with the mass of the host dark matter (DM) halo, where haloes
less massive than ~1012 Mcircledot typically host one dominant galaxy
each while more massive haloes tend to host groups and clusters
of luminous galaxies, as quantified by the halo occupation distribution
 (HOD, Yan, Madgwick & White 2003; Kravtsov et al. 2004;
Abazajian et al. 2005, in 2dF, SDSS and in simulations). The environment
 dependence thus implies that the galaxy properties are
correlated with the host halo mass, with the bimodality at M crit lessorsimilar
1012 Mcircledot (Blanton et al. 2004).
(x) Hot halo gas. Ellipticals of L B > 1010.5 Lcircledot show a significant
 excess of X-ray flux plausibly associated with hot halo gas
(Ciotti et al. 1991; Mathews & Brighenti 2003). Intergalactic X-
ray radiation is detected predominantly in groups where the brightest
 galaxy is an elliptical. Group properties have a transition near
sigma v ~ 140 km s-1 (Helsdon & Ponman 2003; Osmond & Ponman
2004).
(xi) Luminosity/mass functions. The stellar mass function has
a 'knee' near M s,crit, where the shallow dn/dM s  proportional to  M -1s on the
faint side turns into an exponential drop. In contrast, the dark
halo mass function is predicted by the standard Lambda cold dark
matter (Lambda1CDM) model to be dn/d M  proportional to  M -1.8 everywhere below
 ~1013 Mcircledot. A match at M s,crit requires a baryonic fraction
M s/M ~ 0.05, indicating gas loss, and associating M s,crit with
M crit similarequal 6 x 1011 Mcircledot. The halo mass-to-light function has a minimum
 near M crit, varying as M/ L  proportional to  M -2/3 and  proportional to  M+1/2 below and
above it, respectively, thus implying increased suppression of star
formation away from the critical mass on both sides (Marinoni &
Hudson 2002; Bell et al. 2003a,b; Yang, Mo & van den Bosch 2003,
in a B-mag sample and in SDSS, 2MASS, 2dF).
(xii) Fundamental line versus plane. A transition is detected
in the galaxy structural scaling relations near M s,crit, e.g. the surface
brightness changes from µs  proportional to  M 0.6s at lower masses to µs ~constant
at the bright end (Kauffmann et al. 2003b, in SDSS). The correlation
below M s,crit is part of the 'fundamental line' relating stellar mass,
radius and velocity over five decades in M s (e.g. Dekel & Woo
2003). The mean metallicity shows a transition near a similar mass
scale from Z ~ M 0.4s to Z ~ constant (Dekel & Woo 2003; Tremonti
et al. 2004, in SDSS and the Local Group).
(xiii) AGNs. Black hole masses are correlated with their host
spheroid properties (e.g. Tremaine et al. 2002). The optical AGN
population, with high accretion rate and SFR, peaks near M s,crit
with little AGN activity at smaller masses, and is associated with
black hole masses lessorsimilar108 Mcircledot (Kauffmann et al. 2003a, in SDSS).
Radio-loud AGNs, uncorrelated with the optical activity and the
SFR, dominate in larger haloes hosting ~108-9 Mcircledot black holes
(G. Kauffmann private communication).
1.2 Key physical processes
The bimodality imprinted on almost every global property of galaxies
 deserves a simple theoretical understanding. We propose that
the main source of the bimodality is the transition from cold flows
to virial shock heating at a critical scale, in concert with feedback
processes and gravitational clustering that emphasize the same characteristic
 scale. We address the cross-talk between these processes,
and integrate them into a scenario which attempts to address simultaneously
 the variety of observed phenomena. The key processes
are:
(i) Cold infall versus hot medium. The thermal behaviour of
the gas as it falls through the halo is qualitatively different below
and above a critical mass scale of M shock lessorsimilar 1012 Mcircledot (Birnboim &
Dekel 2003; Keres et al. 2005). In less massive haloes, the disc is
built by cold flows (~104-5 K), which are likely to generate early
bursts of star formation. In more massive haloes, the infalling gas is
first heated by stable shocks to near the virial temperature (~106 K).
Near and above M shock at z greaterorequalslant 2 (and preferentially in isolated galax-
ies), streams of dense cold gas penetrate through the dilute shockheated
 medium (Fardal et al. 2001; Kravtsov 2003; Keres et al. 2005)
(discussed in Sections 2-4).
(ii) Gravitational clustering. Non-linear gravitational clustering
 of the DM occurs on a characteristic mass scale, M *, marking
the typical haloes forming at a given epoch and the lower bound

for groups of galaxies. We point out that the clustering scale, which
varies rapidly with cosmological time, happens to coincide with
M shock at z ~ 1, and the interplay between these scales plays a role
in determining the galaxy properties (Section 4).
(iii) Feedback. We argue that the feedback processes affecting
galaxy evolution are relatively ineffective near M shock, largely due
to the shock-heating process itself, and they therefore help emphasizing
 the imprint of this scale on the galaxy properties. Supernova
and other feedback processes regulate star formation in the blue sequence
 below M shock. Feedback by AGNs, or other sources, becomes
efficient in haloes more massive than M shock, because it preferentially
 affects the dilute shock-heated medium and may prevent it
from ever cooling and forming stars effects (Section 5).
We show how the introduction of shock-stability physics crystallizes
 our understanding of the origin of the characteristic scales of
galaxies. We argue that the combination of shock heating, feedback
and clustering introduces a new feature in galaxy formation modelling
 - a complete suppression of cold gas supply in haloes above a
critical mass after a critical redshift. This could be the key to solving
many of the open questions posed by the observations, focusing on
the bright-end truncation of the luminosity function, the appearance
of very red bright galaxies already at z ~ 1 at the expense of big
blue galaxies, and the indications for massive starbursts at higher
redshifts. We note that some of the issues have been addressed in
parallel, in a similar spirit and in different ways, by Birnboim &
Dekel (2003), Benson et al. (2003), Keres et al. (2005) and Binney
(2004). We make here a more thorough investigation into the cold
flows versus shock-heating phenomenon, relate it to the feedback
and clustering processes, and attempt an integrated scenario that
addresses simultaneously the variety of observed features.
1.3 Outline
In Section 2, we provide an improved presentation of the original
analysis of spherical shock stability (Birnboim & Dekel 2003, hereafter
 BD03). In Section 3, we compute the associated critical mass
scale in the cosmological context. In Section 4, we describe how
the phenomena is demonstrated in cosmological simulations, and
learn about cold filaments in massive hot haloes at high redshift.
In Section 5, we address the cross-talk with the relevant feedback
processes working alternatively below and above the critical scale.
Then, in Section 6, we integrate the above processes into a scenario
which attempts to explain the origin of the bimodality and related
features, and report first results from simulations that implement
the new ingredients. In Section 7, we briefly discuss possible implications
 on other open issues in galaxy formation, and in Section 8
we summarize our results, the proposed re-engineering of galaxy
formation simulations, and the open theoretical issues.
2 SPHERICAL SHOCK-STABILITY ANALYSIS
The standard paradigm of disc formation (Rees & Ostriker 1977;
Silk 1977; White & Rees 1978; Blumenthal et al. 1984; White &
Frenk 1991; Mo, Mao & White 1998), which lies at the basis of all
current models of galaxy formation, assumes that while a DM halo
relaxes to a virial equilibrium, the gas that falls in within it is shock
heated near the halo virial radius Rv to the halo virial temperature.
It is then assumed to cool radiatively from the inside out. As long as
the cooling time is shorter than a certain global free-fall time (or the
Hubble time), typically inside a current 'cooling radius', the gas is
assumed to accrete gradually on to a central disc and then form stars
in a quiescent way. The maximum halo mass for efficient cooling
was estimated to be of the order of ~1012-13 Mcircledot, and the common
 wisdom has been since then that this explains the upper bound
for disc galaxies. However, early hints, based on one-dimensional
simulations, indicated that this scenario cannot reproduce the sharp
drop in the luminosity function above this scale (Thoul & Weinberg
 1995). Even earlier studies, valid in the context of the pancake
picture of galaxy formation, indicated that virial shock heating may
not be as general as assumed (Binney 1977). More advanced cosmological
 simulations have started to reveal the presence of cold
flows (Fardal et al. 2001). With the new data from big surveys such
as SDSS, 2MASS and 2dF, and the detailed semi-analytic modelling
 (SAM) of galaxy formation, it is becoming clear that the
observed scale is somewhat smaller and the drop is sharper than
predicted by the original picture. It seems that the current models
based on the standard paradigm have hard time trying to reproduce
many of the observed bimodality features summarized in Section 1.
This motivated us to attempt a closer look at the shock-heating
mechanism.
2.1 Spherical simulations
Fig. 1 shows the time evolution of the radii of Lagrangian gas shells
in a spherical gravitating system consisting of gas (in this case with

primordial composition) and DM, similar to the original simulations
 by BD03 using an accurate one-dimensional hydrodynamical
code. Not shown are the dissipationless DM shells, which detach
from the cosmological expansion, collapse and then oscillate into
virial equilibrium such that they deepen the potential well attracting
 the dissipating gas shells. The initial density perturbation was
assumed to have a profile proportional to the linear two-point correlation
 function of Lambda1CDM (justified in Dekel 1981), and the final
density profile roughly mimics the universal NFW profile (Navarro,
Frenk & White 1997) seen in cosmological simulations. The gas is
cooling radiatively based on the atomic cooling function computed
by Sutherland & Dopita (1993). The collapse of each gas shell is
stopped at roughly 0.05 Rv by an artificial centrifugal force which
mimics the formation of a central disc.
The upper panel focuses on massive haloes of ~1012 Mcircledot.Asex-
pected in the common picture, a strong shock exists near the virial
radius, namely at roughly half the maximum-expansion radius of
the corresponding shell. The virial shock gradually propagates outwards,
 encompassing more mass in time. The hot post-shock gas
is in a quasi-static equilibrium, pressure supported against gravitational
 collapse. The lower panel focuses on halo masses smaller by
an order of magnitude, and shows an interesting new phenomenon.
A stable shock forms and inflates from the disc towards the virial
radius only after a total mass of more than a few times 1011 Mcircledot has
collapsed. In less massive systems, the cooling rate is faster than
the compression rate required for restoring the pressure in the postshock
 gas; had there been a shock, the post-shock gas would have
become unstable against radial gravitational contraction and unable
to support the shock. In the specific case shown, with zero metallicity,
 the critical mass is biased low; with more realistic metallicities
it becomes ~1012 Mcircledot due to the more efficient cooling via metal
lines (Section 3).
We demonstrate below (Section 4) that the behaviour seen in
the spherical simulations is robustly reproduced in cosmological
simulations. But first we wish to understand the spherical case, and
use it for simple predictions.
2.2 Post-shock stability criterion
The existence or absence of a shock, as seen in the simulations, can
be evaluated via a straightforward stability analysis of the post-shock
gas (first introduced in BD03). We provide here a brief, improved
presentation of this analysis, followed by a more detailed estimate
of the predicted critical mass for shock stability as a function of
redshift (Section 3).
A stable extended shock can exist when the pressure in the postshock
 gas is sufficient to balance the gravitational attraction towards
the halo centre. The standard equation of state for an ideal gas expresses
 the pressure as P = (gamma - 1)rho e where rho and e are the
gas density and internal energy per unit mass, and gamma = 5/3 for a
monatomic gas. In the textbook case of no cooling, the adiabatic
index is defined as gamma <=> (\u2202ln P/\u2202ln rho)ad, and the system is known
to be gravitationally stable once gamma>4/3. When there is energy loss
(e.g. by radiation) at a rate q per unit mass, we define a new quantity
along the particle trajectories:
gammaeff <=>dlnrho = gamma -rhoe . (1)
The second equality follows from energy conservation, e =-P V -
q (where V = 1/rho), plugged into the equation of state. Note that
gamma eff = gamma when q = 0. The difference between the two is a ratio of
characteristic rates for the two competing processes: the cooling,
which reduces the pressure in the post-shock gas, and the compression
 due to the pattern of the post-shock infall, which tends
to increase the pressure. If the compression rate is efficient compared
 to the cooling-loss rate, it restores the pressure necessary for
supporting a stable extended shock, but otherwise the post-shock
gas collapses inwards under gravity, failing to support the extended
shock.
It is convenient to express the compression rate in the post-shock
region as the inverse of a compression time, which we define by
tcomp <=> Gamma1rho Gamma1 <=>gamma (3gamma - 4) =5 , (2)
with the factor Gamma1 to be justified below, and the last equality referring
to gamma = 5/3. For a spherical shock at radius rs, and a post-shock
radial velocity u1, we assume that the radial flow pattern in the postshock
 region is homologous, u/r = u 1/r s. This is justified based
on the spherical simulations described above, where the log-linear
post-shock flow lines in Fig. 1 are nearly parallel straight lines. We
then obtain using continuity
tcomp =(-\u2207 · u) =(-3 u
1)
. (3)
The competing cooling rate in the post-shock region is expressed
as the inverse of the standard radiative cooling time defined by
tcool <=>q , (4)
where e = e(T) and q  proportional to  rhoLambda1(T, Z), functions of temperature T and
metallicity Z. Then in equation (1)
gammaeff = gamma - Gamma1t
. (5)
In order to test for stability, BD03 performed a perturbation analysis
 where the radius of a shell is perturbed by r  ->  r + deltar and
the sign of the force, deltar/deltar , is computed. Writing deltar = u deltat, using
the homology, and assuming further that the gravity and pressure
forces balance each other near the transition state between stability
and instability, rho-1 \u2207 P = GM/r 2, one obtains a restoring force, i.e.
stability, for
gammaeff >gammacrit <=>gamma + 2/3 =7 . (6)
The gamma crit = 10/7 replaces the standard gamma crit = 4/3 of the adiabatic
case.2
Using equation (5) and the definitions of the time-scales above,
the shock stability criterion of equation (6) becomes the simple condition
 that the cooling rate should be slower than the compression
rate:
tcool > tcomp. (7)
Once the cooling rate is slower, the pressure gain by compression
can balance the loss by radiative cooling, which allows the postshock
 gas to be stable against global gravitational collapse and thus
support the shock. The factor Gamma1 = 21/5 has been introduced in
the definition of tcomp, equation (2), in order to simplify this final
expression.

Note that the relevant quantity for stability is the ratio of rates
associated with the two competing processes, independent of how
slow each of them actually is in absolute terms. Each of the characteristic
 times could in principle be longer than the Hubble time - it
is their ratio which determines whether a stable shock is possible or
the gas falls in subject to gravity, cold and unperturbed.
2.3 Pre-shock quantities
2.3.1 Compression rate
Using the standard jump conditions across a strong shock, we can
express the characteristic time-scales (or gamma eff) in terms of the preshock
 gas quantities. The jump condition for the radial velocity is
u0 - us =gamma - 1 (u1 - us), (8)
where us is the radial shock velocity and u0 is the radial velocity of
the pre-shock gas. Then
tcomp =3(gamma - 1)|u
0|
parenleftbigg
1 -(gamma - 1)|u
0|
parenrightbigg-1
(9)
=5|u
0|
(1 - 3 us) (10)
similarequal 5.48 Gyr|u
0|
(1 - 3 us), (11)
where us <=> us/|u0| (see Section 2.3.4) and the last expression assumes
 gamma = 5/3, rs in 100 kpc, and u0 in 100 km s-1.Ifus = 0, say,
then tcomp is about six times larger than r s/|u 0|, a typical free-fall
time from rs into the halo centre. At the virial radius, tcomp is comparable
 to the Hubble time at the corresponding epoch, but at inner
radii it becomes significantly shorter.
2.3.2 Cooling rate
The cooling time (e.g. based on Sutherland & Dopita 1993) is
tcool <=>q =1 + epsilon12 kT
bracketleftbigg chi 2
m
bracketrightbigg-1
, (12)
where Lambda1(T, Z) is the cooling function, k is the Boltzmann constant,
epsilon1 <=> n He/n H, the mass per particle is m <=> µm p with µ = (1 + 4epsilon1)/
(2 + 3epsilon1), and the number of electrons per particle is chi = (1 + 2epsilon1)/
(2 + 3epsilon1). For 25 per cent He in mass, one has epsilon1 = 1/12, yielding
µ = 0.59. If we express the post-shock temperature as
T 6 <=> T /106 K, the post-shock baryon density as rho -28 <=>
rho/10-28 gcm-3, and the cooling function as Lambda1-22(T , Z ) <=> Lambda1(T ,
Z )/10-22 erg cm3 s-1,wehave
tcool similarequal 2.61 Gyr rho-28 T6 Lambda1-22(T , Z ). (13)
The post-shock gas density is related to the pre-shock density by
the jump condition
rho1 =gamma - 1 rho0 = 4rho0, (14)
and the post-shock temperature entering the cooling time is related
to the pre-shock radial velocity u0 via
kT1
m
2(gamma - 1)
(gamma + 1)2
2 =
16
2. (15)
We note in passing that for a virial shock, where u 0 =-V v
(BD03), the post-shock temperature is actually
T1 greaterorsimilar8 Tv, (16)
comparable to but somewhat smaller than the virial temperature as
defined in equation (A9).
2.3.3 Stability criterion
Using equations (11) and (13), the critical stability condition becomes

tcool
tcomp
rho-28 T6 Lambda1-22(T , Z )
rs |u0|-1 (1 - 3 us)-1
with rs in 100 kpc and u0 in 100 km s-1. Recall that equation (14) relates
 rho to rho 0, and equation (15) relates T to u0. Thus, for given shock
radius rs, shock velocity relative to infall u s/|u 0|, and pre-shock gas
density rho 0, once the metallicity Z is given and the cooling function
Lambda1(T, Z) is known, one can solve equation (17) for the critical values
of T and the corresponding u0. When put in a cosmological context
(Section 3), this solution is associated with a unique critical halo
mass.
The stability criterion derived above, equation (6) or equation (7),
is found to work very well when compared to the results of the
spherical simulations shown in Fig. 1. When gamma eff (or t cool/t comp)
is computed using pre-shock quantities at a position just outside
the 'disc', we find that as long as the halo is less massive than
a critical scale, before the shock forms, the value of gamma eff is indeed
well below gamma crit and is gradually rising, reaching gamma crit almost exactly
when the shock starts propagating outwards. The value of the gamma eff
computed using the quantities just outside the shock then oscillates
about gamma crit with a decreasing amplitude, following the oscillations in
the shock radius seen in Fig. 1. As the shock eventually settles at the
virial radius, gamma eff approaches 5/3, larger than gamma crit = 10/7, where
the cooling is negligible. The same stability criterion is found to
be valid to a good approximation also in cosmological simulations
(Section 4).
2.3.4 Shock velocity
What value of us is relevant for evaluating stability? In the inner
halo, we use us = 0. This is because, as the halo is growing in mass,
the shock first forms in the inner halo and then propagates outwards
(Fig. 1b). The onset of shock stability is therefore marked by its
ability to develop a velocity outwards.
During the stable phase when the shock is expanding with the
virial radius, the spherical simulations indicate roughly u1 similarequal-us
(Fig. 1a), namely us similarequal 1/7 (equation 8). This indicates that a small
shock velocity of such a magnitude is appropriate for evaluating
stability at the virial radius.
Note that stability is harder to achieve when the shock is expanding
 relatively fast. In particular, in the extreme case us = 1/3, the
post-shock velocity vanishes, u1 = 0 (equation 8). The compression
rate becomes infinitely slow (equation 3), implying that the shock
cannot be stabilized.
3 SHOCK-HEATING SCALE IN COSMOLOGY
3.1 Haloes in cosmology
We wish to translate the critical stability condition, equation (7)
or equation (17), into a critical post-shock temperature, and the

corresponding critical halo virial velocity and mass as a function
of redshift. Equation (17) has a unique solution when combined
with the two virial relations between halo mass, velocity and radius
(equation A6), and the relation between post-shock temperature and
pre-shock infalling velocity (equation 15).
As summarized in Appendix A, the time dependence of the virial
relations can be expressed in terms of the convenient parameter
A <=>Delta1200Omega1m0.3h0.7 a, (18)
where a <=> 1/(1 + z) is the cosmological expansion factor and the
other parameters are of order unity. The parameters Omega1m0.3 and h0.7
correspond to today's values of the cosmological mass density parameter
 and the Hubble expansion parameter, respectively, and for
the standard Lambda1CDM cosmology adopted here they are both equal to
unity. The parameter Delta1200 is the virial density factor given approximately
 in equation (A8): at redshifts z > 1itisDelta1200 similarequal 1, but at
lower redshifts it becomes somewhat larger, reaching Delta1200 similarequal 1.7 at
z = 0.
3.2 Compression rate
For a shock at the virial radius, rs = Rv, we approximate u 0 =-V v,
as predicted by the spherical collapse model in an Einstein-de Sitter
cosmology (BD03, appendix B).
When the shock is at an arbitrary inner radius r, where the infalling
velocity is |u|, we multiply Rv and V v by appropriate factors fr <=>
r/R v and fu <=>|u|/V v (discussed in Section 3.5). Then equation (11)
becomes
tcomp similarequal 14.3 Gyr A fr fu (1 - 3 us). (19)
3.3 Cooling rate: gas density
In order to express the cooling time of equation (13) in terms of cosmological
 quantities, we first evaluate the pre-shock baryon density,
which we write as
rhob = 4 fb (rho/¯rho)vir Delta1rhou frho. (20)
Here, rho u is the universal mean mass density (equation A4), and Delta1 is
the top-hat mean overdensity inside the virial radius (equation A8).
The factor (rho/¯rho)vir translates ¯rho, the mean total density interior to
Rv,torho, the local total density at Rv. The effective baryonic fraction
f b turns it into a pre-shock baryonic density. The factor 4 stands for
the ratio between the post-shock gas density and the pre-shock gas
density.3 The factor f rho <=> rho(r )/rho(R v) reflects the ratio of the actual
gas density at some radius r within the halo to its value at the virial
radius (see Section 3.5).
The ratio rho/¯rho at the virial radius is derived for the universal NFW
halo density profile revealed by cosmological simulations (Navarro,
Frenk & White 1997). For a virial concentration parameter c, this
ratio isparenleftbigg
rho
¯rho
parenrightbigg
=
3(1 + c)2
bracketleftbigg
ln(1 + c) -(1 + c)
bracketrightbigg-1
. (21)
A typical concentration of c = 12 is associated with rho/¯rho similarequal 0.17;
we therefore express the approximate results below using the factor
f ¯rho,0.17 <=> (rho/¯rho)/0.17. In our more accurate evaluation of the critical
scale (Section 3.7), we model the dependence of the mean concentration
 on mass and time using the fit of Bullock et al. (2001) for
the Lambda1CDM cosmology:
c(M, a) = 18 M11 a. (22)
The effective baryon fraction f b may in principle be as large as
the universal fraction similarequal 0.13, but it is likely to be smaller because of
gas loss due to outflows. For the approximate expressions we define
f b,0.05 <=> f b/0.05.
The gas density at r, equation (20), thus becomes
rho-28 = 0.190 A fb,0.05 frho f ¯rho,0.17. (23)
Inserting this baryon density into equation (13), the cooling time
becomes
tcool = 13.7 A fb,0.05 frho f¯rho,0.17 T6 Lambda1-22(T , Z ) Gyr. (24)
The cooling function that we use below (based on Sutherland &
Dopita 1993) neglects two physical processes: Compton scattering
off the cosmic microwave background and the possible effect of
external radiation on the cooling rate through the re-ionization of
hydrogen. Based on the more complete cooling function as implemented
 by Kravtsov & Gnedin (2004), one learns that these
processes become important only for densities below ~10-28 and
10-26 gcm-3 at z ~ 0 and 4, respectively. Using equation (23), we
conclude that while these processes may have a certain effect on
the cooling rate near the virial radius, they should be negligible
once the analysis is applied inside the inner half of the halo, where
the critical scale for shock heating is determined in practice. We
address these effects in more detail elsewhere (Birnboim, Dekel &
Loeb in preparation).
3.4 Metallicity
The metallicity near the virial radius and in the inner halo, which
also enters the cooling rate, is one of our most uncertain inputs. For
the mean metallicity Z (in solar units) as a function of redshift z,we
use the two-parameter functional form
log(Z /Z0) =-sz, (25)
where Z 0 is today's metallicity and the slope s governs the rate of
growth.
An upper limit may be imposed by the hot, X-ray emitting intracluster
 medium (ICM) at low redshifts, which indicate Z 0 ~ 0.2-0.3.
The ICM metallicity evolution in SAMs, assuming a range of different
 feedback recipes, yields consistently an average enrichment
rate of roughly s similarequal 0.17 (R. Somerville private communication; De
Lucia, Kauffmann & White 2004). We adopt this enrichment rate s
in our modelling below.
A realistic estimate of the metallicity near the virial radius (or
perhaps a lower limit for the inner halo) may be provided by C IV
absorbers in the intergalactic medium at z ~ 2-4 (Schaye et al.
2003). At densities appropriate to typical NFW haloes at z = 3
(with c = 3), namely rho vir similarequal 53rho u, they measure an average of
[C/H] =-2.47. Silicon measurements, on the other hand, seem
to indicate a metallicity that is about five times larger (A. Aguirre
private communication). If one takes the geometrical mean between
the metallicities indicated by C IV and by Si one has Z(z = 3) similarequal
0.0075. This translates to Z 0 = 0.025 if s = 0.17.
We note that another popular indicator, Mg II, indicates consistently
 Z < 0.01 within 50-100 kpc of galaxy centres at z ~ 1 (private
communication with J. Charlton; e.g. Ding et al. 2003).

The damped Lyalpha systems (DLAS) are believed to sample cold gas
deeper inside the haloes, and can thus provide another interesting
limit. Observations in the range z = 1-4 (Prochaska et al. 2003) can
be fitted on average by equation (25) with Z 0 similarequal 0.2 and a somewhat
steeper slope s similarequal 0.26. However, a fit with s = 0.17 (and then Z 0 =
0.1) is not ruled out.
Based on the above estimates, we adopt as our fiducial metallicities
 Z 0 = 0.03 at Rv and Z 0 = 0.1 at the 'disc' radius ~0.1R v, both
with an enrichment rate s = 0.17.
3.5 Inside the halo
For a shock in the inner halo we wish to estimate the factors fr, fu
and f rho .
Empirically from the spherical simulation of BD03, for a shell
encompassing a mass just shy of the critical mass (as well as from
the toy model of BD03 of gas contracting in a static isothermal
sphere), we estimate for fr <=> r/R v
frho <=>rho
0(Rv)
similarequal
braceleftbigg f -1.6
f -2.1r , r ~ 0.1Rv
We adopt below f rho = f -2r at fr = 0.1.
Energy conservation assuming pure radial motion inside a static
singular isothermal sphere yields
fu <=>u
0(Rv)
=1 + 2 fb fr - 1 + 2(1 - fb)ln fr . (27)
For fr = 0.1 and f b = 0.05 this gives the estimate fu similarequal 2.5.
Based on the definition of fu, the temperature behind a virial shock
is related to the temperature obtained from the stability condition at
radius r by
T (Rv) = fu T (r ). (28)
3.6 Crude explicit estimates
The critical temperature for stability, as obtained by comparing tcool
and tcomp in the cosmological context, equations (24) and (19), is
T6 lambda-22(T , Z ) = 1.04 A F, (29)
where
F <=> fr fu frho fb,0.05 f ¯rho,0.17(1 - 3 us). (30)
The cooling function as computed by Sutherland & Dopita (1993)
can be crudely approximated in the range 0.1 < T 6 < 10 by
Lambda1-22 similarequal 0.12 Z0.03 T6 + 0.02 T6 , (31)
where Z 0.03 <=> Z /0.03, with Z in solar units. The above expression is
valid for -2.5 lessorequalslant log Z lessorequalslant 0, and at lower metallicities the value of Lambda1
is practically the same as for log Z =-2.5. This fit is good near T 6
~ 1 for all values of Z. The first term refers to atomic cooling, while
the second term is due to bremsstrahlung. For an approximation
relevant in haloes near M shock we ignore the bremsstrahlung term,
which becomes noticeable only at higher temperatures. One can
then obtain in equation (29) an analytic estimate for the critical
temperature:
T6 similarequal 0.36 AZ0.03 F. (32)
Using equations (15) and (28), with |u 0|=V v just outside the
virial radius, we then obtain for the critical velocity and mass
V100 similarequal 1.62 AZ0.03 F fu (1 + us) , (33)
M11 similarequal 25.9 AZ0.03 F fu (1 + us). (34)
A comment regarding the us dependence of our results. The critical
 temperature depends on the shock velocity us via F, T  proportional to 
(1-3 us)-1/2, reflecting the us dependence of tcomp. The critical temperature
 is thus monotonically increasing with us. An additional us
dependence enters when the temperature is translated to a critical
virial velocity using the jump condition, V  proportional to  (1 + us)-1 T 1/2, and
then to a critical mass, M  proportional to  (1 + us)-3 T 3/2. For a slowly moving
shock, us lessmuch 1/3, the combined us dependence of the critical mass
is M  proportional to  [1+(9/4) us](1-3 us) similarequal 1-(3/4) us - a decreasing function
of us. This means that at a given radius in a halo of a given mass,
when everything else is equal, a slowly expanding shock is actually
more stable than a shock at rest. For example, if the shock is expanding
 with us = 1/7 rather than us = 0, the critical mass is smaller by
about 24 per cent. However, recall that stability is harder to achieve
when the shock is expanding relatively fast, and the compression
completely vanishes if us greaterorequalslant 1/3 (Section 2.3.4).
For actual crude estimates of the critical scales at z = 0, we assume
fb,0.05 similarequal f ¯rho,0.17 similarequal 1. For a shock at the virial radius, fr = fu =
f rho = 1, we assume Z 0 similarequal 0.03 and us similarequal 1/7, and obtain
T6 similarequal 0.5, V100 similarequal 1.6, M11 similarequal 26. (35)
At an inner radius closer to the disc vicinity, say fr = 0.1, we
estimate fu similarequal 2.5 and f rho similarequal 100 (Section 3.5). Assuming Z 0 similarequal 0.1
and us similarequal 0, we obtain
T6 similarequal 1.1, V100 similarequal 1.1, M11 similarequal 8.8. (36)
We see that for a shock at r ~ 0.1R v, the expected critical mass is
smaller than at Rv, somewhat below ~1012 Mcircledot.
The above estimates are useful for exploring the qualitative dependences
 of the critical values on redshift, metallicity and gas fraction.
For example, in equation (34), the explicit redshift dependence and
the decrease of metallicity with redshift tend to lower the critical
mass towards higher z. On the other hand, the decrease of halo concentration
 with z (i.e. increase of f ¯rho,0.17), and the possible increase
of the effective gas fraction with z (Section 6), tend to push the
critical mass up at higher z.
3.7 More accurate estimates
We now obtain a better estimate of the critical temperature (and then
critical mass and virial velocity) by solving equation (29) using the
exact cooling function of SD93 and adopting specific models for
the time evolution of metallicity and halo structure. The results are
presented graphically.
The baryon density is computed assuming an NFW profile whose
concentration evolves in time as in equation (22). The effective fraction
 of cold gas is assumed to be f b = 0.05, motivated by best fits
of SAMs to the Milky Way (Klypin, Zhao & Somerville 2002)
and by fitting the Lambda1CDM halo mass function to the observed luminosity
 function near L* (Bell et al. 2003a). The metallicity evolution
 is parametrized as in equation (25) with s = 0.17 for to-
day's metallicities in the range Z 0 = 0.03-0.3. Upper and lower
estimates for the critical scales are evaluated at the virial radius
and at r = 0.1R v, respectively, using the crude estimates of Section
 3.5. In the following figures the shock is assumed to be at
rest, u s = 0.
Fig. 2 shows the critical mass as a function of redshift. At a typical
inner halo radius, r = 0.1 R v, we plot the curves for three different
current metallicities: Z 0 = 0.03, 0.1, 0.3. The critical halo mass,

for Z 0 = 0.1, is similarequal6 x 1011 Mcircledot quite independent of redshift. The
uncertain metallicity introduces a scatter by a factor of 2 up and
down (for z < 2.5).
An upper limit of ~2 x 1012 Mcircledot is obtained for a shock at Rv
when a correspondingly low metallicity is assumed, Z 0 = 0.03.
When the assumption of us similarequal 0 is replaced by us similarequal 1/7, allowing
the shock to expand with the virial radius as seen in Fig. 1(a), the
critical mass at Rv with Z 0 = 0.03 becomes comparable to that at
0.1 R v with Z 0 = 0.3.
Fig. 3 shows the corresponding virial velocity. At z = 0, the critical
virial velocity for a shock in the inner halo is ~120 km s-1, with a
±30 km s-1 scatter due to metallicity. The critical virial velocity
increases monotonically with redshift, to ~200 km s-1 near z ~ 3
(a crude fit to the redshift dependence is V v = 120 + 28z).
The dependence on metallicity at z = 0 is highlighted in Fig. 4. The
metallicity enters strongly through the cooling function Lambda1(T, Z). The
critical mass grows roughly like Z 1/2, as predicted in equation (34),
so it spans about an order of magnitude over the whole metallicity
range.
The analytic estimates of equations (33) and (34), based on the
approximate cooling function, turn out to provide good estimates
in most cases, and can therefore be used for extending the results
analytically to any desired choice of the relevant parameters.
We learn that the critical halo mass for shock stability at the disc
vicinity, M shock(r disc), is somewhat smaller than for a shock at the
virial radius, M shock(R v). This result is robust: it is true even if the
metallicity at the virial radius is smaller by an order of magnitude
than the metallicity at the disc, and even when us at Rv is as large as
1/7. This means that as the halo is growing in mass, the conditions
for a stable shock develop first in the inner halo and somewhat later
in the outer halo. Thus, in haloes of mass M < M shock(r disc), we
expect cold flows with no shock heating throughout the halo. In
the other extreme of haloes of mass M >M shock(R v), we expect
shock heating of most of the gas by a shock near the virial radius.
In haloes of mass in the narrow intermediate range M shock(r disc) <
M < M shock(R v), we expect shock heating somewhere inside the
halo, preventing most of the gas from falling in and giving rise to a
hot medium. This predicted mass range, of a factor of 2-3 in mass,
is consistent with the range of transition from all cold to mostly hot
seen in cosmological simulations (Section 4).
Also shown in Fig. 2 are the typical masses of haloes forming at
different redshifts, the 1sigma (termed M *) and 2sigma halo masses according
 to the Press-Schechter formalism, equation (A18). According
to the improved Sheth-Tormen version, the corresponding fractions
of the total mass encompassed in haloes exceeding the mass M are
22 and 4.7 per cent, respectively. One can see in Fig. 2 that M shock
coincides with M * at z ~ 1, and with the 2sigma mass at z ~ 3.4. By

z ~ 2, say, most of the forming haloes are significantly less massive
than M shock. When embedded in a large-scale high sigma -density peak,
the distribution of forming haloes at a given z may shift towards more
massive haloes. In fact, the most massive halo in a volume of comoving
 size ~100 Mpc is likely to be more massive than 1012 Mcircledot
at all relevant redshifts (z < 6, say). Nevertheless, the qualitative
result concerning the majority of the haloes remains valid. We conclude
 that in the vast majority of forming galaxies the gas has never
been shock heated to the virial temperature - it rather flows cold all
the way to the disc vicinity.
We note that the values obtained for M shock at low redshifts are
compatible with the observed bimodality/transition scale summarized
 in Section 1. The estimates in the inner halo, using the lower
and upper limits for Z 0, indeed border the observed characteristic
halo mass of ~6 x 1011 Mcircledot. The upper-limit estimate at Rv corresponds
 to a halo mass similar to that of the Milky Way.
4 COLD STREAMS IN HOT HALOES
4.1 Cosmological simulations
Cosmological hydro simulations indicate that the phenomenon of
cold flows is a general phenomenon not restricted to spherical symmetry.
 Fig. 5 displays snapshots of an Eulerian simulation from
Birnboim et al. (in preparation).4 Shown are maps of gas temperature
 in two epochs in the evolution of a protogalaxy: one at z similarequal
4, when the halo is already relative massive, and the other at z similarequal
9, when the halo is still rather small. While the more massive halo,
near the critical scale, shows a hot gas component near the virial
temperature behind a virial shock, the smaller halo shows only cold
flows inside the virial radius.
Similar results have been obtained earlier from smoothed particle
hydrodynamics (SPH) simulations by Fardal et al. (2001), who emphasized
 the feeding of galaxies by cold flows preferentially at early
epochs. Based on our spherical analysis, we understand that this
redshift dependence mostly reflects the smaller masses of haloes at
higher redshifts. Keres et al. (2005) have analysed similar SPH simulations
 and presented the case for the two modes of infall, cold and
hot, in more detail.5 For example, in their figs 1 and 2 they demonstrate
 the two-mode phenomenon very convincingly by showing the
distribution of particles and their trajectories in temperature-density
diagrams. Their most informative fig. 6 shows the fractions of cold
and hot infall as a function of halo mass at different redshifts. For all
haloes of masses below a critical mass the infall is predominantly
cold. Near the critical mass there is a relatively sharp transition into
a hot mode, which becomes dominant above the critical mass. The
transition from 100 per cent cold to more then 50 per cent hot occurs
across a range of only a factor ~2 in halo mass. In this simulation,
where zero metallicity is assumed, the transition mass is M ~ 3 x
1011 Mcircledot at all redshifts in the range 0-3. A similar transition as a
function of halo mass, and the constancy of the critical mass as a
function of redshift, are both reproduced in the Eulerian cosmological
 simulations studies in Birnboim et al. (in preparation).
The spherical simulations described above (Section 2.1), and the
corresponding analytic analysis (Sections 2 and 3), yield very similar
 results. In fact, our analytic predictions for the case of zero
metallicity match the critical mass measured by Keres et al. (2005)
remarkably well.
The stability criterion derived in the spherical case turns out to
be valid locally in the cosmological simulations where the nonspherical
 features are pronounced. Birnboim et al. (in preparation)
use this criterion to identify the cold streams and hot media in the
simulations without explicit information concerning the presence or
absence of actual shocks. When testing the criterion in these simulations,
 in which the hot and cold phases may be present in the same
halo, the local gas properties at each position is first transformed to
post-shock quantities, as if there was a shock there, and the stability
is evaluated based on the derived value of gamma eff there. The resultant
 maps of gamma eff resemble quite well the temperature maps of the

actual simulation. This demonstrates that the wisdom gained by the
spherical analysis is applicable in the general case.
4.2 Filaments in the simulations
The cosmological simulation of Fig. 5 also reveal that massive
haloes at high redshift can have cold streams embedded in the hot
medium. While the hot medium is rather spherical, the cold flows are
filamentary and sometimes clumpy. Fig. 6 highlights the filamentary
 nature of the cold gas in the same big halo. This phenomenon is
consistent with the findings of Keres et al. (2005) that the cold mode
may in some cases coexist with the hot mode even above M shock, especially
 at z > 2 (their fig. 6) and preferentially in relatively isolated
galaxies. We wish to understand the origin of this phenomenon, and
learn about its dependence on cosmological time.
The simulation results of Keres et al. (2005) provide several additional
 relevant clues. First, their figs 17 and 18 indicate that the
cold infall indeed tends to be filamentary, especially at high redshifts,
 while the hot mode is more spherical. They report that the
directional signal measuring filamentary infall in the cold accretion
mode is stronger for haloes above M shock while the infall is more
isotropic below it.
Secondly, Keres et al. (2005) display in their fig. 13 the environment
 dependence of the gas-infalling modes, showing that the cold
and hot modes dominate at low and high neighbourhood densities,
respectively. We learn from this figure that at z < 2 the cold mode
dominates for galaxy densities below n gal ~0.3 (h-1 Mpc)-3 and becomes
 negligible at larger environment densities, while at z = 3 the
cold mode is more pronounced than the hot mode for all neighbourhood
 densities up to n gal ~ 10 (h-1 Mpc)-3. The correlation between
the environment density and the host halo mass implies that this z
variation of the environment dependence could be partly attributed
to the finding of a significant residual cold mode in massive haloes
at high z (e.g. their fig. 6).
Thirdly, Keres et al. (2005) show in their fig. 16 that the cold
accretion is on average of higher density than the hot mode. This is by
only a factor of ~2 (probably underestimated because they mix small
and large haloes), but since the shock is responsible for a density
increase by a factor of 4, the actual overdensity of the hypothetical
post-shock (or pre-shock) gas is more like ~8. Similarly, Nagai
& Kravtsov (2003) find in their simulation of a massive halo that
the filamentary structure is associated with gas entropy ( proportional to  T /rho2/3)
far below that of the surrounding halo gas. In the high-resolution
simulation shown in Fig. 6, we find that the density in the cold
streams is actually higher than the surrounding gas density by two
orders of magnitude or more. The higher gas density in the filaments
is associated with a more efficient cooling which prevents a shock
from forming along the filaments (Section 2).
We find that the cold gas filaments at the halo outskirts are strongly
correlated with the DM filaments that feed this halo (reported in detail
 in Seleson & Dekel in preparation). These filaments are part
of the large-scale cosmic web. They enter massive haloes at high
redshift as narrow streams with a density higher than the halo average
 by a factor of a few. The initial overdensity of the gas flowing
along the DM filaments scales with the DM density, while its inflow
velocity is comparable to the halo virial velocity. As a result, the
initial cooling time in the thin filaments is shorter by a factor of a
few than in the surrounding spherical halo, while the compression
time is comparable in the filaments and the host halo. Equation (7)
then implies that the thin filaments have a harder time supporting a
stable shock. The gas filaments remain cold, and become denser as
the stream penetrates through the hot medium into the halo centre.
The result is that in massive haloes at high redshift the critical halo
mass for shock heating in the filaments feeding them is larger than
the estimate for a spherical virial shock derived in Section 3. We
provide below a crude estimate for this revised critical mass.
4.3 Interplay with the clustering scale
What is the reason for the appearance of cold streams in massive
haloes above M shock at high z? First, recall that there is another characteristic
 scale in the problem - the scale of non-linear clustering
M *, determined by the shape and amplitude of the initial fluctuation
 power spectrum and its growth rate. The masses for 1sigma haloes
(M *) and 2sigma haloes, based on equation (A18) with nu = 1 and 2,
respectively, are shown again in Fig. 7. We note that M shock ~ M *
at z lessorequalslant 1, while M shock greatermuch M * at z > 2. This means that ~1012 Mcircledot
haloes are typical at z < 1 but they are the highest rare peaks at
z > 2. We argue that this is responsible for the difference in the coldfilament
 behaviour of M greaterorsimilar M shock haloes in the two epochs. Since
the large-scale structure of DM is roughly self-similar in time (when
measured in terms of M * and the background universal density), we
can learn about the difference between typical and rare haloes by
comparing M ~ M * and M greatermuch M * haloes in a single simulation
snapshot. One can see in any high-resolution cosmological N-body
simulation (e.g. the 'Millennium Run', visualized in www.mpa-
garching.mpg.de/galform/millennium) that the rare massive haloes
tend to be nodes fed by a few intersecting relatively narrow filaments
 which are denser than the virial density of these haloes. On
the other hand, a typical M * halo is commonly embedded in a single
 filament of the cosmic web, and this halo thus sees a wide-angle
inflow pattern in which the matter density is comparable to the virial
density [this is quantified in Seleson & Dekel (in preparation)]. This
explains why greaterorsimilar1012 Mcircledot haloes are fed by narrow dense streams at
z > 2 but not at z < 1.
A crude way to estimate the maximum halo mass for cold streams
at a given redshift is as follows. Recall that the critical ratio for shock

stability (equation 17) scales roughly like
tcool
tcomp
rho-1 T Lambda1-1
RV-1
with T, R and V the halo virial quantities and rho the gas density. At a
given epoch, the typical halo rho and R/V are roughly independent of
halo mass (based on the definition of the virial radius), so with the
virial relation T  proportional to  M 2/3, and approximating the cooling function
with Lambda1  proportional to  T -1 (equation 31), the critical ratio for spherical infall in
virialized haloes isparenleftbigg
tcool
tcomp
parenrightbigg
=
parenleftbigg M
Mshock
parenrightbigg4/3
. (38)
The analogous critical ratio in the dense streams inside a halo of
mass M, assuming that RV -1 in the streams is the same as in the halo,
is inversely proportional to the density enhancement rho stream/rho halo
(equation 37). Our estimates from N-body simulations indicate that
rho stream/rho vir ~ (fM */M)-2/3 with f ~3 (Seleson & Dekel in prepa-
ration). With equation (37) one obtains
parenleftbigg t
tcomp
parenrightbigg
=
parenleftbigg fM
M
parenrightbigg2/3 parenleftbigg M
Mshock
parenrightbigg4/3
. (39)
For this ratio to equal unity in the streams, the critical halo mass is
Mstream ~fM
Mshock, fM* < Mshock. (40)
This maximum mass for cold streams is shown in Fig. 7. At low z,
where fM * > M shock, cold streams exist only for M < M shock.At
high z, where fM * < M shock, cold streams appear even in M >M shock
haloes where shocks heat part of the gas, as long as M < M stream.
The critical redshift zcrit separating these two regimes is defined by
fM*(zcrit) = Mshock. (41)
This scenario is consistent with the cosmological hydrodynamical
 simulations. The shock-heating mass explains the transition
from cold to hot at a given mass roughly independent of z, and the
presence of cold streams above M shock at z > z crit explains the dependence
 of the cold mode on redshift and environment. Besides
its dependence on halo mass, the environment effect (e.g. Keres
et al. 2005) may also be due to the survivability of cold streams in
different environments. While streams could survive unperturbed
in relatively isolated galaxies, they are likely to be harassed by the
active intergalactic environment in dense groups. The environment
dependence may therefore also reflect variations in the HOD at a
given halo mass. The properties of cold flows in haloes as a function
of halo mass, redshift and grouping deserve a detailed analysis using
high-resolution cosmological hydro simulations.
5 FEEDBACK AND LONG-TERM SHUTDOWN
Once the halo gas is shock heated in massive haloes, what is the
process that keeps it hot and maintains the shutdown required by
the bimodality? Is it also responsible for the rise of M/L with mass
above M s,crit (and the absence of cooling flows in clusters)? Several
feedback mechanisms can heat the gas. We suggest that they have
a minimum effectiveness in haloes ~M shock. This can be largely
induced by the shock heating itself, and in turn it can amplify the
bimodality features. Some of the feedback mechanisms are limited
to smaller haloes, while others, such as AGN feedback, are likely to
be important in more massive haloes. The latter can be triggered by
the shock heating and then help maintaining the gas hot for a long
time.
5.1 Below the shock-heating scale
(i) Supernova feedback. Based on the physics of supernova
(SN) remnants, the energy fed to the gas in haloes of T ~ 105 Kis
proportional to the stellar mass despite significant radiative losses
(Dekel & Silk 1986). When compared to the energy required for
significantly heating the gas, one obtains a maximum halo virial
velocity for SN feedback, V SN similarequal 120 km s-1. This is only weakly
dependent on the gas fraction, density or metallicity (Dekel & Silk
1986, equation 49), and is therefore insensitive to redshift. Only
in potential wells shallower than V SN can the SN feedback significantly
 suppress further star formation and regulate the process. Fig. 3
shows V SN and Fig. 2 shows the corresponding mass versus redshift.
With an effective f b ~ 0.05, the corresponding stellar mass at z =
0is~3.5 x 1010 Mcircledot, practically coinciding with the bimodality
scale. The similarity of the SN and shock-heating scales is partly a
coincidence, because the nuclear origin of the initial SN energy has
little to do with galactic cooling or dynamics. However, there is an
obvious similarity in the cooling processes and in the asymptotic
behaviour of an SN remnant, which is not a strong function of its
initial energy. The distinct correlations between the properties of
galaxies below M s,crit indeed point at SN feedback as its primary
driver. These correlations define a 'fundamental line', V  proportional to  M 0.2s ,
Z  proportional to  M 0.4s , µ  proportional to  M 0.6s , where µ is surface brightness (Kauffmann
et al. 2003b; Tremonti et al. 2004). SN feedback can explain the
origin of the fundamental line (Dekel & Woo 2003) based on (1) the
above energy criterion, which implies M s/M  proportional to  V 2; (2) the virial
relations (equation A6); (3) the instantaneous recycling approximation,
 Z  proportional to  M s/M gas and (4) angular momentum conservation, R *  proportional to 
lambdaR, with lambda a constant spin parameter (Fall & Efstathiou 1980).
(ii) UV-on-dust feedback. Also working below M shock are
momentum-driven winds due to radiation pressure on dust grains,

arising from the continuum absorption and scattering of ultraviolet
 (UV) photons emitted by starbursts or AGNs Murray, Quataert
& Thompson (2005). The dust survives and can provide sufficient
optical depth if the gas is cold and dense, e.g. in the cold flows
below M shock, which can also provide the starbursts responsible for
the required UV flux and metals. Since dust grains cannot survive
above ~106 K, M shock imposes an upper bound for this feedback.
(iii) Photoionization feedback. The UV radiation from stars
and AGNs ionizes most of the gas after z ~ 10 (Bullock, Kravtsov
& Weinberg 2000; Loeb & Barkana 2001), heats it to greaterorsimilar104 K and
prevents it from falling into haloes below the Jeans scale V v ~
30 km s-1 (Gnedin 2000). As the ionization persists for cosmological
 epochs, the hot gas evaporates via steady winds from haloes
smaller than a similar scale (Shaviv & Dekel 2003). While this is
important in dwarf galaxies, it cannot be very relevant to the bimodality
 at M shock.
5.2 Above the shock-heating scale
(i) AGN feedback. The fact that AGNs exist preferentially in
haloes above M crit may be due to a lower limit for haloes hosting
black holes (Koushiappas, Bullock & Dekel 2004), to starvation of
AGNs by SN feedback in haloes below M crit, or to another reason.
The power emitted from AGNs, e.g. in their radio jets, seems to be
more than necessary for keeping the gas hot. Given a black hole
mass M BH ~107 Mcircledot V 4100 in a galaxy of velocity dispersion V, and
assuming that a fraction epsilon1 of this mass is radiated away, the ratio of
energies is E AGN/E gas ~ 7 x103 epsilon1 fb,0.05 V100.Forepsilon1>10-3, there
seems to be enough AGN energy for affecting most of the halo
gas. If this energy is released during relatively quiet, long phases
of self-regulated AGN activity, it can keep the gas hot. However,
black hole physics does not seem to explain the characteristic scale
of bimodality. Furthermore, had the energy ratio been a measure of
feedback effectiveness, it would have implied a decline with mass,
in conflict with the trend of M/L.
The shock heating of the gas into a dilute medium is likely to make
it vulnerable to heating and pushing by the central energy source,
thus providing the trigger for effective AGN feedback. Simulations
of winds in a two-phase medium demonstrate that the dilute phase is
pushed away while the dense clouds are hardly affected (Slyz et al.
2005). This behaviour is likely to be generic, though the mechanism
by which the energy released near the black hole is spread in the
halo gas is an open issue (Ruszkowski, Bruggen & Begelman 2004;
Scannapieco & Oh 2004). If so, the feedback efficiency may be
driven by the relative fraction of hot gas rather than the actual AGN
energy. Fig. 6 of Keres et al. (2004) shows that the hot fraction varies
roughly  proportional to M 1/2, implying M/L  proportional to  M 1/2 near and above M shock,in
qualitative agreement with the observed trend (Section 1, item k).
In this scenario, the shutdown scale arises naturally from the shock
heating.
(ii) Two-phase medium. Given that the cooling function peaks
near greaterorsimilar104 K, the virialized gas at greaterorsimilar106 K develops a two-phase
medium, with cold, dense clouds pressure confined within the hot,
dilute medium (Field 1965; Fall & Rees 1985). The cloud sizes and
evolution are affected by thermal conductivity and dynamical processes
 (Voigt & Fabian 2004). This can help explaining the brightend
 truncation of the blue sequence (Maller & Bullock 2004). Some
of the gas is locked in the orbiting clouds and the density rho hot of the
hot gas is reduced, slowing the cooling and the infall. Approximating
 equation (17) with t cool/t comp  proportional to  rho-1 T 2v, and recalling that T v
 proportional to  M 2/3v , the longer cooling time makes the critical mass for further
shock heating (t cool/t comp ~ 1) smaller by a similar factor, M shock  proportional to 
rhohot , namely it enhances the shock stability. The gas may be kept hot
over longer periods by repeating shocks due to continuous accretion
into the halo, which may alleviate the need for AGN feedback. Still,
a necessary condition for hot gas is the initial shock heating, i.e.
being in a halo above M shock.
(iii) Dynamical-friction feedback. Another heating source is
the dynamical friction acting on galaxies in a halo core. The energy
transferred is comparable to that required for preventing cooling
flows in cluster centres (El-Zant, Kim & Kamionkowski 2004b).
The gas response to dynamical friction, unlike the DM response,
has a sharp peak near a Mach number of unity (Ostriker 1999, fig.
3), namely when the gas is heated to near the virial temperature in
M > M shock haloes and not in smaller haloes hosting cooler gas. As
groups occur above a critical halo mass that roughly coincides with
M shock at z lessorequalslant 1, this feedback appears almost simultaneously with
the hot medium, which then serves as the vulnerable victim of the
same feedback (as for AGN feedback).
Fig. 8 is an illustration of the strength of the different feedback
processes, referring to the gas fraction that could have been heated at
z ~ 0. The figure highlights the fact that different processes dominate
below and above M shock lessorsimilar 1012 Mcircledot. The transition from cold to hot
infall has a crucial role in determining the feedback efficiencies near
the critical mass; it induces a minimum in feedback efficiency at a
critical scale M fdbk ~M shock and drives the shapes of the curves
about this minimum. At higher redshifts this minimum becomes
wider and deeper but centred on a similar critical mass.
6 THE ORIGIN OF BIMODALITY
6.1 A scenario from the assumed ingredients
We propose that the cold flows and shock heating play a key role
in producing the observed bimodality features. These features are
emphasized by the similarity between the scales associated with
shock heating, feedback and clustering. Based on our current understanding
 of these physical processes, we assume the validity of
the following.
(i) A new mode of star formation. The collisions of the (partly
clumpy) cold streams with each other and with the inner disc are

assumed to produce starbursts, analogous to collisions of cold
gaseous discs or clouds. These collisions are expected to produce
isothermal shocks, behind which the rapid cooling generates dense,
cold slabs where the Jeans mass is small. While the details are yet
to be worked out, we assume that such a mode of star formation
may be responsible for much of the stars in the universe. In some
cases, this can be an enhanced quiescent mode, leaving the disc
intact without producing a big spheroid, an in other cases it may
resemble the starbursts associated with mergers.
(ii) Hot forever. Once the gas in a massive halo is shock heated
to near the virial temperature, it is assumed to be hot forever. This is
based on the slow cooling time of the dilute hot medium and its vulnerability
 to AGN feedback, while cold, dense clouds and streams
could be better shielded against winds and ionizing radiation. The
shock heating is thus assumed to trigger a shutdown of all modes
of star formation in haloes where cold streams do not prevail.
(iii) Cold streams in a hot medium. Cold streams in haloes
above M shock (Section 4) are assumed to supply cold gas for further
disc growth and star formation, preferentially before z crit ~ 2 and in
isolated galaxies. After zcrit, especially in groups, cold streams are
suppressed and a complete shutdown of star formation is assumed
to follow.
The bimodality in colour (or stellar age or SFR) versus mass, the
correlation with the environment and the evolution with redshift,
all emerge naturally from the early efficient star formation followed
after z ~ 2bytheabrupt shutdown in haloes above M shock, which
typically host groups. This is illustrated in Fig. 9.
(i) The blue sequence. It is dominated by galaxies in haloes
below M shock, as they grow by accretion/mergers. Cold flows lead
to early disc growth and star formation, which is regulated by SN
feedback over cosmological times. Galaxies can get very blue because
 of repeating starbursts due to the clumpy gas supply and the
interplay between infall, starburst and outflow.
(ii) Bright blue extension. Some galaxies continue to be fed by
cold streams even when they are more massive than M shock, extending
 the blue sequence beyond L*. This occurs especially at z > z crit ~
2, when the streams feeding high-sigma haloes are relatively narrow and
dense, resulting in massive starbursts.
(iii) The red sequence. Once a halo is more massive than M shock,
halo gas is shock heated; it becomes dilute and vulnerable to AGN
feedback. At z < z crit, where cold streams are suppressed, gas supply
from the host halo shuts off, preventing any further growth of discs
and star formation. If residual cold gas has been consumed in earlier
mergers, there is a total shutdown of all modes of star formation
above M shock, allowing the stellar population to passively turn 'red
and dead' into the red sequence. The massive tip of the blue sequence
at z > z crit becomes the massive tip of the red sequence at z < z crit,
explaining the bright-end truncation of the blue sequence at low z,
and the appearance of brighter, very red galaxies already at z ~ 1.
Subsequent growth along the red sequence is induced by gas-poor
mergers uncontaminated by new blue stars.
(iv) Faint red extension. When colour is plotted against stellar
mass, the bimodality extends to smaller galaxies which are typically
satellites of the central galaxies in common haloes. In haloes below
M shock, accretion on to satellite galaxies can keep them on the blue
sequence for a while. In haloes above M shock, where gas supply stops
and the environment density is high, the satellites too become red
and dead.
In an associated paper (Cattaneo et al. 2006, hereafter C05) we
have implemented the proposed new physics in a hybrid SAM/N-
body simulation (GalICS; Hatton et al. 2003). The main new ingredient
 is a shutdown in gas cooling and star formation above M shock
~ 1012 Mcircledot after z crit ~ 3, while allowing for efficient star formation
 by cold streams in more massive haloes prior to zcrit. The
revision yields excellent fits to the observed features at low and
high redshifts (C05). Fig. 10 shows one example of the results -
colour-magnitude diagrams which demonstrate the success of this
model along the lines envisioned in Fig. 9. The top panel highlights
the main deficiencies of the 'standard' model at z = 0: an excess
of bright blue galaxies accompanied by a shortage of red-enough
galaxies compared to the SDSS data (Baldry et al. 2004). The 'new'
model puts the blue and red sequences where they should be at
z = 0, with a proper truncation at the bright-blue end and appropriately
 red colours in the red sequence. The z = 3 diagram shows the
predicted bright blue galaxies (which were absent in the 'standard'
model, C05). The colours distinguish between galaxies in haloes
below (blue) and above 2 x 1012 Mcircledot, comparable to the mass separating
 haloes hosting field galaxies and groups. This indicates that

the model recovers the correlation between colour and environment
density. It also recovers the bimodality in bulge-to-disc ratio (C05).
6.2 Origin of observed features
The colour-magnitude bimodality emerges from the colour-mass
bimodality lying at the basis of the model. The red satellites extend
the luminosity range over which the blue and red sequences coexist
and highlight the gap in colour. This gap is amplified because the
galaxies making the transition into the red sequence once their haloes
become >M shock tend to be the merger remnants with big bulges and
AGNs from the red tip of the blue sequence.
The correlated bimodality in bulge-to-disc ratio can be understood
 in the scenario where most of the big spheroids in the red
sequence are passively aged galaxies that have grown massive stellar
 components already in the blue sequence (C05). The transition to
the red sequence is likely to be made by galaxies with big spheroids
because (1) they have consumed their gas in the same mergers that
produced the spheroids, (2) these mergers tend to occur in big haloes
hosting groups where shock heating stops the gas supply and (3)
these spheroids contain massive black holes that can keep the gas hot.
The strong anticorrelation of SFR (and blue colour) with the number
 density in the environment is a natural outcome of the mass
dependence. The key is that the minimum halo mass for groups
at ~1012 Mcircledot is comparable to M shock at z lessorequalslant 1. The strong dependence
 of cold gas supply on host halo mass can therefore be
responsible for the distinction between the SFR in field and clustered
 galaxies (which is thus predicted to be limited to late times).
The galaxies dominating low-HOD haloes below M shock enjoy cold
gas supply leading to discs forming stars. The galaxies populating
 groups of subhaloes, typically in haloes above M shock at z lessorequalslant
1, suffered starvation of external gas supply and lost their internal
gas in mergers, thus stopped forming stars and passively evolved
to the red sequence. The faint end of the red sequence, which is
preferentially present in high environment densities, is due to the
starvation of satellite galaxies in the high-HOD haloes above M shock.
The model naturally predicts the secondary bimodality seen along
the red sequence (along the lines of Berlind et al. 2005).
The classic morphology-environment relation, traditionally attributed
 to the correlation of merger rate with the environment, may
also be viewed as a result of the cold-flow phenomenon. New discs
are predicted to form only in haloes below M shock, namely in 'field'
galaxies, and not in the group haloes above M shock. On the other hand,
the frequent major mergers in groups help build the big spheroids
preferentially there. The abrupt shutdown of star formation above
M shock makes the colour-environment correlation stronger than the
morphology-environment correlation.
The environment dependence highlights an interesting cross-talk
between the clustering and gas processes. The distinction between
haloes hosting a single dominant galaxy and haloes hosting groups
has traditionally been attributed to gas cooling on a dynamical timescale
 (Rees & Ostriker 1977). Our shock-stability analysis helps
quantifying this idea. However, it seems from N-body simulations
that the gravitational DM clustering process has a parallel role
(Kravtsov et al. 2004). The HOD of subhaloes develops a transition
from single to multiple occupancy near a comparable halo mass, associated
 with the current M *. The HOD of subhaloes is similar to the
HOD of galaxies deduced from the observed correlation function
for galaxies (Section 1, item ix). One can conclude that in haloes
above ~1012 Mcircledot the potential wells associated with the DM subhaloes
 provide the sites for the fragmented gas collapse on the scales
preferred by cooling, thus emphasizing this scale as the minimum
scale for groups. The coincidence between these scales is behind
the environment dependence of the bimodality features. We have
discussed above the other possible role of M * in the appearance of
cold filaments in hot haloes at high z (Section 4) and in some of the
feedback processes (Section 5).
The minimum in mean halo mass-to-light ratio M/L near
~1011-12 Mcircledot can be attributed to the maximum in gas supply
near M shock dictated by the shock heating above this mass and the
associated minimum in feedback there (Fig. 8). While SN feedback
gets stronger towards smaller haloes, AGN feedback pumps energy
more effectively into the shock-heated medium in more massive
haloes. The 'fundamental line' due to SN feedback below M shock
corresponds to M/L  proportional to  M -2/3 (Dekel & Woo 2003), while the
transition from cold to hot infall indicates M/L  proportional to  M 1/2 above

M shock, in the ballpark of the findings from 2dF (Section 1, item k).
This settles the discrepancy between the halo mass function and the
galaxy luminosity function both below and above the bimodality
scale. Our model predicts a similar behaviour of M/L(M)atz ~
1, to be revealed by spectroscopic surveys (such as the DEEP2/3
Madgwick et al. 2003a; Coil et al. 2004), which have already
confirmed the predicted invariance of the bimodality mass in this
redshift range (Fig. 7).
The massive starbursts at high z, primarily due to cold streams,
are helped by an increase in M shock itself. At high z, the upper bound
for SN feedback becomes lessmuchM shock (Fig. 2), allowing for a more
efficient gas accretion at lessorsimilarM shock.Iff b is 0.13 instead of 0.05 at
z ~ 3, the value of M shock doubles (equation 34). Another factor
proportional to f b enters when translating from halo to stellar mass,
yielding a total increase of ~5 in the critical stellar mass at high z.
We thus expect a strong star formation activity at high z in galaxies
with stellar masses exceeding ~1011 Mcircledot.
The global star formation history could be derived from the predicted
 SFR as a function of mass and redshift, convolved with the
time evolution of the halo mass function in the given cosmology.
With the prediction that haloes lessorsimilarM shock are the most efficient star
formers, and with the Press-Schechter estimate that haloes of such
a mass typically form at z ~ 1 (Fig. 2), the star formation density
is predicted to peak near z ~ 1, with a relatively flat behaviour towards
 higher redshifts and a sharp drop towards lower redshifts, as
observed. In particular, the cumulative stellar density seems to stop
growing quite abruptly at z ~ 1 (Dickinson et al. 2003), when the
typical forming haloes become larger than M shock and the cold flows
are suppressed. The growth of the hot fraction as a function of halo
mass (Keres et al. 2005) can be translated to the drop in SFR after
z ~ 1. The 'downsizing' of star formation in galaxies is also helped
by the shutdown of star formation above M shock while smaller galaxies
 can make stars also later, until they fall into bigger haloes.
7 DISCUSSION: OTHER IMPLICATIONS
Possible implications on open issues in galaxy formation where
further study is desired are worth mentioning.
X-rays. The shock-heating scale may be detectable in soft X-
rays, as the lower limit for galaxies and groups containing hot halo
gas. In turn, the suppression of heating below M shock explains the
missing soft X-ray background (Pen 1999; Benson et al. 2000). We
predict a noticeable suppression of X-ray emission in the range 5 x
105-2 x 106 K.
Lyalpha emission. The cold (~104 K) flows may instead be an efficient
 source of Lyalpha radiation, possibly associated with observed
Lyalpha emitters at high redshift (Kurk et al. 2003). It has been argued
 based on SPH simulations (Fardal et al. 2001; Furlanetto et al.
2003) that the flows radiate their infalling energy mostly in Lyalpha
before they blend smoothly into the discs. Our Eulerian simulations
(Zinger et al. in preparation) indicate that the streams do eventually
shock in the inner halo. This produces X-ray, but, given the high
density there, the X-ray radiation is likely to be confined to an ionized
 Stromgren sphere of a few kiloparsecs. This energy eventually
transforms into Lyalpha radiation, which could propagate out via thermal
 broadening and systematic redshifts. A study involving radiative
transfer is required.
Damped Lyalpha systems. The possible association of massive cold
flows with DLAS (Prochaska et al. 2003) should be addressed in
cosmological simulations.
LIRGS. Cold flows may explain the massive starbursts associated
with LIRGs at z lessorsimilar 1 (Hammer et al. 2005). If half the stars in today's
discs were formed in such LIRGs, and the majority of galaxies
today are fragile discs, than many of the LIRGs could not have been
produced by violent major mergers. The cold streams may provide a
less violent starburst mechanism not associated with the destruction
of discs. Simulations that properly incorporate cold streams should
be confronted with these data.
Angular momentum. The proposed scenario may set the stage
for solving the angular momentum puzzle - the overproduction of
low angular momentum spheroids in current cosmological simulations
 (Navarro & Steinmetz 2000). The solution should involve the
removal of baryons with low angular momentum. In small galaxies,
 SN feedback can blow the gas away from their small building
blocks, which are otherwise the main source of low angular momentum
 via many minor mergers (Maller & Dekel 2002; Maller,
Dekel & Somerville 2002). In galaxies near M shock, we find from
cosmological simulations (Zinger et al. in preparation; see Fig. 6)
that the low angular momentum gas is typically associated with
the shock-heated medium, which can be prevented from cooling
by AGN feedback. The cold streams come from larger distances
with ~50 per cent higher specific angular momentum, appropriate
 for producing extended discs. The feedback effects, both below
and above the critical scale, have not been properly simulated yet
because of incomplete treatment of the microphysics.
Cold clouds. The formation of discs by a clumpy cold gas phase
may have the following implications. (1) It may explain the starbursts
 responsible for very blue galaxies. (2) It may help explaining
the bright-end truncation of the luminosity function (Section 5). (3)
The dynamical friction bringing the clouds into the disc transfers
energy into the halo, which may help explaining the discrepancy between
 the steep cusps predicted by N-body simulations and the flat
cores indicated by rotation curves in low-surface-brightness galaxies
 (Dekel et al. 2003a; Dekel, Devor & Hetzroni 2003b; El-Zant
et al. 2004a; Ma & Boylan-Kolchin 2004). (4) The same process
may lower the predicted maximum rotation velocity in discs at a
given luminosity, balancing the adiabatic contraction of the dark
halo, and thus repair the zero-point offset in models of the Tully-
Fisher relation (e.g. Klypin et al. 2002; Abadi et al. 2003; Dutton
et al. 2005). (5) This may explain the lack of anticorrelation between
the residuals in velocity and radius at a given luminosity (Courteau
& Rix 1999), indicating comparable contributions of disc and dark
halo to the gravitational potential at the effective disc radius (Dutton
et al., in preparation).
Dust lane. Edge-on discs above V v similarequal 120 km s-1 show a welldefined
 dust lane, while less massive discs show diffuse dust above
and below the disc (Dalcanton, Yoachim & Bernstein 2004). A thick,
turbulent, dusty gas phase is indeed expected when SN feedback
is effective, and when cold streams shock and produce stars, both
predicted below an appropriate scale.
Shock heating in dwarf haloes. The cold/hot infall and feedback
 processes are expected to give rise to two scales characterizing
 dwarf galaxies. The lower bound at V v ~ 10-15 km s-1
(e.g. Dekel & Woo 2003, Fig. 3) is commonly attributed to the
drop in atomic cooling rate below ~104 K. We propose that this involves
 shock heating, in analogy to M shock discussed above. Fig. 11
(bottom) shows the quantity relevant to shock stability, t cool/t comp
(equation 17), as a function of halo mass, now stretched to low
masses. The molecular hydrogen cooling rate, relevant below 104 K,
is weaker and may actually be eliminated after z ~ 10 due to
molecule dissociation by the UV background (Haiman, Rees &
Loeb 1996). The stability is evaluated at z = 0 both in the disc
vicinity (r = 0.1Rv, Z = 0.1, us = 0, and near the virial radius
(Z = 0.03, us = 1/7).

The top panel illustrates the deduced evolution of shock radius
between the disc and the virial radius as the halo mass grows in time.
Haloes below ~109 Mcircledot have stable virial shocks. As the halo grows
to ~6 x 108 Mcircledot, the conditions near the disc become unfavourable
for a stable shock, but the virial shock persists. Only when the
halo becomes ~6 x 109 Mcircledot shock heating stops completely. Thus,
the shock heating prevents star formation in smaller haloes even
before the epoch of re-ionization. Once the first stars form in the
dense centres of big clouds, their UV radiation re-ionizes the gas,
dissociates molecules and helps preventing cooling. This provides a
natural explanation for the absence of luminous galaxies with haloes
below ~109 Mcircledot, predicting a large population of mini haloes with
V v < 10 km s-1 which are completely dark. There is only a narrow
window, 6 x 109-6 x 1011 Mcircledot, for haloes that allow cold flows and
can form luminous discs at low redshifts. This range is somewhat
broader at high redshifts, with the lower bound dropping below
109 Mcircledot and the upper bound rising above 1012 Mcircledot prior to z ~ 3.
Within this window, it seems that below V v ~ 30 km s-1 most of
the haloes are totally dark6 and the others are populated by gas-poor
dwarf spheroidals. This is the scale predicted by photoionization
feedback (Section 5).
8 CONCLUSION
8.1 Summary of results
The classic argument of cooling on a dynamical time-scale (Rees
& Ostriker 1977; White & Rees 1978), with order-of-magnitude
estimates of the time-scales involved, provided an inspiring qualitative
 upper bound for luminous galaxies, at a halo mass of M ~
1012-13 Mcircledot. An analytic study of the actual shock-heating process
(Birnboim & Dekel 2003, and this paper) now yields a more concrete
 halo critical scale at M similarequal 6 x 1011 Mcircledot, somewhat smaller
than the original estimate. The criterion for critical shock stability,
tcool = tcomp, (42)
is a balance between the cooling rate and the post-shock compression
 rate, which restores the pressure supporting the shock against
gravitational collapse. The compression time is somewhat larger
than the crossing time at the shock position. The absolute magnitudes
 of these time-scales are irrelevant - they could in principle
both be longer than the Hubble time, because what matters for shock
heating versus cold flows is only the relative rates of the competing
processes. The most relevant critical scale is obtained in the inner
halo, because as the halo grows, the shock first becomes stable in
the inner halo, and it then propagates outwards to the virial radius.
Haloes of mass below the threshold mass build discs in their centres
by cold flows, while in haloes above the threshold much of the gas
is shock heated. These results are confirmed by spherical hydrodynamical
 simulations. The same phenomenon is seen at a comparable
scale in general cosmological hydrodynamical simulations. They reveal
 that in haloes near the critical scale, and even in larger haloes
preferentially at z greaterorequalslant 2 and especially in field galaxies, cold streams
along the filaments feeding the galaxy penetrate through the hot
medium, and allow further disc growth and star formation.
The interplay between these cold flows and shock heating, the
gravitational clustering scale, and the different feedback processes
acting below and above a similar mass scale, is proposed to be
responsible for the robust bimodality imprinted on the observed
galaxy properties. Cold flows in haloes much bigger than the clustering
 scale allow massive starbursts at z greaterorequalslant 2, while shock heating
in comparable haloes at later times shuts off star formation and
leads to big red galaxies. While SN and radiative feedbacks regulate
 star formation below the critical scale, the presence of dilute,
shock-heated gas in more massive haloes allows the AGN feedback
(or another energetic source capable of affecting big galaxies) to
keep the shock-heated gas hot and prevent further disc growth and
star formation. The observed bimodality and many of the related
phenomena are argued to arise naturally from such a scenario (Sec-
tion 6). The shock-heating process also plays a role in introducing
a lower bound to haloes hosting galaxies, at ~109 Mcircledot. The mass
range where disc galaxies can form today turns out to be quite narrow,
 between a few times 109 Mcircledot to slightly below 1012 Mcircledot.
8.2 Re-engineering of SAMs
Once the new physical processes are properly incorporated in the
detailed models of galaxy formation, they solve many of the apparent
conflicts between theory and observation. At a crude level, one might
have naively thought that since the cooling time is anyway assumed
to be shorter in smaller haloes, the details of the cold flows and
shock heating would not matter much to the final result. However,
a closer inspection reveals that there are several key features which
make a qualitative difference as stated below.
(i) Star formation. The near-supersonic cold streams provide a
new efficient mechanism for early star formation. This is in contrast
to the gradual infall of cooling shock-heated gas assumed in most
SAMs, which starts from near rest, accretes smoothly into the disc,
and joins the quiescent mode of star formation there. We find that

the expected cold-gas supply is significantly more efficient than
assumed in most current models even in small haloes (Cattaneo
et al. in preparation).
(ii) Heating inside out. The concept of an expanding 'cooling
radius' used in current SAMs is limited to massive haloes where a
virial shock exists. Otherwise, it is the shock causing the heating
which is propagating from the inside out.
(iii) Shutdown of star formation. The combination of shock
heating and AGN feedback provides a mechanism for shutting off
disc growth and star formation above a threshold halo mass.
(iv) Cold streams. Cold streams that penetrate through the hot
media continue to make discs and produce stars in haloes above
M shock. This happens mostly at z greaterorequalslant 2, and preferentially in less
grouped galaxies, allowing big blue galaxies mostly at high z and
some at low-density environments, and enforcing a sharp shutdown
of star formation at late times and especially in clustered galaxies.
A practical schematic recipe for the critical halo mass below
which cold streams prevail and above which one may assume a
shutdown of gas supply and star formation is
Mcrit =
braceleftBigg
Mshock, z < zcrit
MshockshockfM*(z) , z > zcrit
where the critical redshift zcrit is defined by fM *(z crit) = M shock, the
clustering scale M *(z) is given by equation (A18), and f is a numerical
 factor of order of a few. Our best estimates for the parameters
are log M s similarequal 11.8 (but possibly another value in the range 11.3-
12.3) and f similarequal 3 (possible range: 1-10). Using the approximation
log M * similarequal 13.1-1.3z (z lessorequalslant 2), we obtain zcrit similarequal 1.4 for f = 3. This
recipe should allow big blue systems at z greaterorequalslant 2, eliminate big blue
systems and make big red galaxies at z lessorequalslant 1, and generate a bimodality
 near M shock. This scheme can be refined to allow for a smooth
transition above the critical scale by applying the shutdown to a
varying fraction of the gas and by breaking the streams into clumps
which will generate high peaks of starbursts.
In addition, one may wish to have an effective minimum requirement
 for the central black hole mass in order to ensure enough
feedback energy for maintaining the gas hot. This may emphasize
the bimodality gap in colour and bulge-to-disc ratio. However, the
proposed shutdown by halo mass may be enough for ensuring sufficient
 bulge mass and black hole mass.
The SAMs should be re-engineered to incorporate these processes
and thus help working out the detailed implications of the proposed
scenario. Preliminary attempts to do that, using two different SAMs,
indicate that the incorporation of the new proposed processes outlined
 above indeed leads to significantly better fits with the observed
bimodality features along the lines proposed in Section 6 (C05).
8.3 Open issues
In parallel, the physics of the involved ingredients should be properly
worked out in more detail, starting with the following two hypotheses
 that were laid out in Section 6.
(i) Fate of cold streams. A detailed investigation is required
of the way by which the cold streams evolve and eventually merge
with the central disc, the associated star formation and the resulting
feedback process. While progress can be made using toy models
and simplified simulations, a proper analysis will require simulations
 of higher resolution than are currently available. In particular,
whether or not the predicted starbursts could be associated with
the big dusty sources indicating massive star formation at high redshifts,
 such as the SCUBA sources (Chapman et al. 2003), remains
to be determined once the theory is worked out and the observed
characteristics of these sources are clarified.
(ii) AGN feedback. The physics of AGN feedback is another
unknown. One wishes to understand how the available energy originating
 near the central black hole is transferred to the hot gas spread
over the halo. The physics of how thermal conductivity may heat
the gas is also to be investigated. The increased efficiency of these
feedback mechanisms in the presence of a hot medium as opposed
to their effect on cold flows and clumps are to be quantified.
Parallel attempts to work out the details of the physical input and
to incorporate it in quantitative models of galaxy formation will lead
to progress in our understanding of the galaxy bimodality and the
associated features.
ACKNOWLEDGMENTS
We thank our collaborators A. Cattaneo, S. M. Faber, A. Kravtsov,
E. Neistein, J. Primack, P. Seleson, R. Somerville and E. Zinger.
We thank R. Dave, N. Katz, D. Keres and D. Weinberg for sharing
with us the results of their simulations. We acknowledge stimulating
discussions with J. Binney, D. Lin, G. Kauffmann, G. Mamon, J. P.
Ostriker and D. Weinberg. This research has been supported by ISF
213/02 and NASA ATP NAG5-8218. AD acknowledges support
from a Miller Visiting Professorship at UC Berkeley, a Visiting
Professorship at UC Santa Cruz and a Blaise Pascal International
Chair by Ecole Normale Superiere at the Institut d'Astrophysique,
Paris.
REFERENCES

APPENDIX A: USEFUL RELATIONS
We summarize here the cosmological relations used in the analysis
of Section 3. This is rather basic material, based for example on
Lahav et al. (1991); Carroll, Press & Turner (1992) and Mo & White
(2002). By specifying it here in a concise and convenient form, we
hope to allow the reader to reproduce our results and use them in
future work. Additional relations associated with the spherical tophat
 collapse model are brought in the appendix of BD03.
A1 Cosmology
The basic parameters characterizing a flat cosmological model in
the matter era are the current values of the mean mass density
parameter Omega1m and the Hubble constant H 0. At the time associated
with expansion factor a = 1/(1 + z), the vacuum-energy density
parameter is Omega1Lambda1(a) = 1 - Omega1m(a) and
Omega1m(a) =
Omega1Lambda1 + Omega1ma-3
The Hubble constant is
H (a) = H0Omega1Lambda1 + Omega1ma, (A2)
and the age of the universe is
tuniv(a) =3 H (a)
-1(|1 - Omega1m(a)|/Omega1m(a))1/2
(|1 - Omega1m(a)|)1/2

The mean mass density is
rhou similarequal 1.88 x 10Omega1ha similarequal 2.76 x 10Omega1m0.3 h0.7 a, (A4)
where Omega1m0.3 <=> Omega1m/0.3, h <=> H 0/100 km s-1 Mpc-1, and h0.7 <=>
h/0.7.
A2 Virial relations
The virial relations between halo mass, velocity and radius,
Vv =R
,(4pi/3)R3
= Delta1rhou (A5)
become
M11 similarequal 6.06 V100 A similarequal 342 R1 A, (A6)
where M 11 <=> M v/1011 Mcircledot, V 100 <=> V v/100 km s-1, R 1 <=>
R v/1 Mpc, and
A <=>Delta1200 Omega1m0.3 h0.7 a. (A7)
An approximation for Delta1(a) in a flat universe (Bryan & Norman
1998) is
Delta1(a) similarequal18pi - 82Omega1Lambda1(a) - 39Omega1Lambda1(a)/Omega1m(a). (A8)
The virial temperature can be defined by
kTv
m
1
2
For an isotropic, isothermal sphere, this equals sigma 2, where sigma is the
one-dimensional velocity dispersion and the internal energy per unit
mass is e = (3/2) sigma 2. Thus
V100 similarequal 2.79 T6 M11 similarequal 28.2 T6 A, (A10)
where T 6 <=> T v/106 K.
A3 Press Schechter
Linear fluctuation growth is given by (Lahav et al. 1991; Carroll
et al. 1992; Mo & White 2002)
D(a) =g(1) a, (A11)
where
g(a) similarequal2 Omega1m(a)
x
bracketleftbigg
Omega1m(a) - Omega1Lambda1(a) +1 + Omega1
Lambda1(a)/70
bracketrightbigg-1
.
(A12)
The CDM power spectrum is approximated by (Bardeen et al.
1986)
P(k)  proportional to  kT(k), (A13)
with
T (k) =2.34 q
x [1 + 3.89q + (16.1q) + (5.46q) + (6.71q)],
(A14)
where
q =parenleftbigOmega1
mhMpc
parenrightbig . (A15)
It is normalized by sigma 8 at R = 8 h-1 Mpc, where
sigma(R) =2pi
integraldisplay \u221E
dkk P(k) W(kR), (A16)
and with the Fourier transform of the top-hat window function
W (x ) = 3(sin x - x cos x )/x 3. (A17)
In the Press Schechter (PS) approximation, the characteristic halo
mass M *(a) is defined as the mass of the 1sigma fluctuation,
1 = nu(M, a) =D(a) sigma (M) ,deltac similarequal 1.69, (A18)
where M and the comoving radius R are related via the universal
density today: M = (4pi/3) ¯rho0 R3. The mass of 2sigma fluctuations is
obtained by setting nu(M, a) = 2, etc. Based on the improved formalism
 of Sheth & Tormen (2002), the fraction of total mass in haloes
of masses exceeding M is
F (> M, a) similarequal 0.4
parenleftbigg
1 +nu0.4
parenrightbigg
erfc
parenleftbigg 0.85 nu
\u221A2
parenrightbigg
. (A19)
This fraction for 1, 2 and 3sigma fluctuations is 22, 4.7 and 0.54 per
cent, respectively.
Fig. 2 shows the PS mass M * as a function of redshift. For the
standard Lambda1CDM with sigma 8 = 0.9, its value at z = 0isM *0 = 1.36 x
1013 Mcircledot. One can see that an excellent practical fit in the range
0 lessorequalslant z lessorequalslant 2 is provided by a power law in this semilog plot: log M * =
13.134 - 1.3z. At larger redshifts this gradually becomes an underestimate.
 Trying to provide crude power-law approximations, we
find that M *  proportional to  a4.2  proportional to  t 3.5 are crude approximations in the range 0
lessorequalslant z lessorequalslant 1, and that M *  proportional to  a5  proportional to  t 4 are good to within a factor of 2
in the range 0 lessorequalslant z lessorequalslant 2. These power laws become overestimates at
higher redshifts.

______________________________________________________________
Racah Institute of Physics, The Hebrew University, Jerusalem, Israel
Accepted 2005 December 16. Received 2005 December 12; in original form 2004 December 13
starE-mail: dekel@phys.huji.ac.il (AD); yuval@phys.huji.ac.il (YB)
1 Quoting only sample references, making no attempt to be complete.
Figure 1. Time evolution of the radii of Lagrangian gas shells (lines) in a
spherical simulation of a protogalaxy consisting of primordial gas (Z = 0)
and DM. Temperature is marked by colour. A shock shows up as a sharp
break in the flow lines, namely a sudden slowdown of the infall, associated
with an abrupt increase in the temperature. The lower discontinuity where
the inflow is brought to a final halt marks the 'disc' radius, formed due to an
artificial centrifugal force. (a) A massive system, where the virialized mass
grows from 1011 to 1013 Mcircledot. (b) A less massive system, growing from
1010 to 1012 Mcircledot. A virial shock exists only in systems more massive than a
critical mass, while in smaller haloes the gas flows cold and unperturbed into
the inner halo. With more realistic metallicities the critical mass becomes
~1012 Mcircledot.
cool
2 If the spherical symmetry assumed above is replaced by planar symmetry,
both for the shock and the gravitational field, the stability criterion gamma eff >
10/7 is replaced by gamma eff > 10/11 (Birnboim et al., in preparation). One
can therefore assume in general that the actual critical value lies somewhere
between these two limits; if gamma eff < 10/11 there is no stable shock, if gamma eff >
10/7 the conditions allow a stable shock, and if 10/11 <gammaeff < 10/7 the
shock stability depends on the local geometry.
2
0s
vir
2
3 In the spherical simulations, the relevant factor relating the baryon density
to the DM density in equation (20) is actually closer to ~3 because of a
'bump' in the DM density just inside the virial radius.
rv
Figure 2. Critical shock-heating halo mass as a function of redshift. The
three solid (red) curves refer to a shock at the inner halo, r = 0.1 Rv, with
different metallicities as indicated; the middle curve (Z 0 = 0.1) is our best
estimate. The dash-dotted (magenta) curve refers to a shock at the virial
radius with Z 0 = 0.03. The other parameters used are: f b = 0.05, u s = 0,
s = 0.17 (see text). Shown for comparison (short dash, blue) are the Press-
Schechter estimates of the forming halo masses, corresponding to 1sigma (M *)
and 2sigma , where the fractions of total mass in more massive haloes are 22 and
4.7 per cent, respectively. Also shown is the critical mass for SN feedback
discussed in Section 5 (long dash, green).
Figure 3. Same as Fig. 2 but for the corresponding halo virial velocity.
Figure 4. Critical shock-heating halo mass as a function of metallicity at
redshift z = 0. The solid (red) curve refers to a shock at the inner halo,
r = 0.1 R v. The dash-dotted (magenta) curve refers to a shock at the virial
radius.
4 A description of the simulation technique can be found in Kravtsov (2003),
where it was used for other purposes.
5 The hot phase becomes an 'infalling' mode in this simulation after the gas
cools, but in reality it may be kept hot and be prevented from falling in by
feedback effects, Section 5.
Figure 5. Snapshots from a cosmological hydrodynamical simulation (Birn-
boim et al. in preparation) showing the gas temperature in a slice of a protogalaxy
 at two different epochs, when it has two different masses. The
temperature is relative to the virial temperature of the halo at that time. The
side of each slice is scaled to be 3R v (numbers in comoving h-1 Mpc). Top:
At z similarequal 4, when the halo is already relatively massive, M similarequal 3 x 1011 Mcircledot.
Much of the gas is heated by a strong shock near the virial radius (white
circle). Cold streams penetrate through the hot medium deep into the halo.
Bottom: At z similarequal 9, when the halo is still small, M similarequal 2 x 1010 Mcircledot. The gas
flows in cold (T lessmuch T v), showing no evidence for shock heating inside the
virial radius (circle).
Figure 6. A snapshot from a cosmological hydrodynamical simulation
(Zinger et al. in preparation) showing the gas density of the cold flows
(T < 104.5 K, rho>10-26.3 gcm-3) within the virial radius of the same massive
 galaxy shown in Fig. 5 at z = 4. The cold phase is filamentary. In the
outer radii, the gas filaments tend to ride on the large-scale DM filaments
feeding the halo. About half the mass of the cold phase is in dense clumps.
Figure 7. Cold streams and shock-heated medium as a function of halo mass
and redshift. The nearly horizontal curve is the typical threshold mass for a
stable shock in the spherical infall from Fig. 2, below which the flows are
predominantly cold and above which a shock-heated medium is present. The
inclined solid curve is the upper limit for cold streams from equation (40)
with f = 3; this upper limit is valid at redshifts higher than z crit ~ 1-2,
defined by M shock > fM *. The hot medium in M > M shock haloes at z >z crit
hosts cold streams which allow disc growth and star formation, while haloes
of a similar mass at z < z crit are all hot, shutting off gas supply and star
formation.
halo
cool
stream
*
*
Figure 8. The 'strength' of the various feedback processes at z = 0, e.g. referring
 to the fraction of the initial gas that has been heated or removed
(schematic). Different feedback processes are effective below and above the
critical shock-heating scale lessorsimilar1012 Mcircledot, and the feedback efficiency is at a
minimum near this scale, giving rise to a minimum in M/L there.
Figure 9. Schematic illustration of the origin of the bimodality in colour
(or SFR or stellar age) as a function of halo mass. At z > 2 (top), continuous
gas supply, regulated by SN feedback, makes the galaxies evolve along the
blue sequence, which extends beyond the shock-heating mass due to cold
streams in hot media. At z < 1 (middle), in the absence of cold streams above
M shock, the shock-heated gas is kept hot by AGN feedback, gas supply and
star formation shut down, and the stellar population passively turns red and
dead. Gas-poor mergers stretch the red sequence towards larger masses.
When the halo mass is replaced by stellar mass (bottom), the red sequence
is stretched towards small stellar masses due to satellite galaxies sharing a
common halo. The colour is correlated with the environment density via the
halo mass, with the minimum group mass being comparable to M shock.
Figure 10. Colour-magnitude diagrams from a hybrid semi-analytic/N-
body simulation (C05) demonstrating the success of the proposed scenario
in reproducing the main bimodality features along the lines of Fig. 9. Top:
Based on the 'standard' version of the GalICS SAM. Middle: The result
of incorporating a shutdown in star formation above M shock ~ 1012 Mcircledot
after z crit ~ 3, and allowing for star formation even in >M shock haloes prior
to zcrit. Bottom: Same at z = 3. Blue dots refer to galaxies in haloes below
M shock, while red and green dots are galaxies in haloes above M shock, central
and satellite galaxies, respectively. The halo mass correlates with number
density in the environment. Contours mark number density of galaxies in
the diagram. The colour curves mark the main bodies of the blue and red
sequences in the SDSS data (Baldry et al. 2004).
Figure 11. Shock stability as a function of mass at z = 0. Bottom: The
ratio of rates, t cool/t comp, as a function of halo mass derived at the disc
(Z = 0.1, us = 0) and at the virial radius (Z = 0.03, us = 0.15). The cooling
rate is assumed to vanish for T < 104 K. A stable shock is possible once
tcool/tcomp > 1. Top: An illustration of the evolution of shock radius between
the disc vicinity and the virial radius as the halo mass grows in time, based
on the stability criterion shown in the bottom panel.
6 This can be deduced from the discrepancy between the flat faint-end luminosity
 function and the steep halo mass function predicted in Lambda1CDM,
given the Tully-Fisher like velocity-luminosity relation of dwarf galaxies
(e.g. Dekel & Woo 2003, fig. 3).
Abadi M. G., Navarro J. F., Steinmetz M., Eke V. R., 2003, ApJ, 591, 499
Abazajian K. et al., 2005, ApJ, 625, 613
Baldry I. K., Glazebrook K., Brinkmann J., Ivezic Z., Lupton R. H., Nichol
R. C., Szalay A. S., 2004, ApJ, 600, 681
Balogh M. et al., 2004, MNRAS, 348, 1355
Bardeen J. M., Bond J. R., Kaiser N., Szalay A. S., 1986, ApJ, 304, 15
Bell E. F., Baugh C. M., Cole S., Frenk C. S., Lacey C. G., 2003a, MNRAS,
343, 343
Bell E. F., McIntosh D. H., Katz N., Weinberg M. D., 2003b, ApJ, 585, L117
Bell E. F., McIntosh D. H., Katz N., Weinberg M. D., 2003c, ApJS, 149, 289
Bell E. F. et al., 2004, ApJ, 608, 752
Benson A. J., Bower R. G., Frenk C. S., White S. D. M., 2000, MNRAS,
314, 557
Benson A. J., Bower R. G., Frenk C. S., Lacey C. G., Baugh C. M., Cole S.,
2003, ApJ, 599, 38
Berlind A. A., Blanton M. R., Hogg D. W., Weinberg D. H., Dav R., Eisenstein
 D. J., Katz N., 2005, ApJ, 629, 625
Binney J., 1977, ApJ, 215, 483
Binney J., 2004, MNRAS, 347, 1093
Birnboim Y., Dekel A., 2003, MNRAS, 345, 349 (BD03)
Blanton M. R., Eisentstein D., Hogg D. W., Zehavi I., 2004, preprint (astro-
ph/0411037)
Blanton M. R., Eisentstein D., Hogg D. W., Schlegel D. J., Brinkmann J.,
2005, ApJ, 629, 143
Blumenthal G. R., Faber S. M., Primack J. R., Rees M. J., 1984, Nat, 311,
517
Bryan G. L., Norman M. L., 1998, ApJ, 495, 80
Bullock J. S., Kravtsov A. V., Weinberg D. H., 2000, ApJ, 539, 517
Bullock J. S., Kolatt T. S., Sigad Y., Somerville R. S., Kravtsov A. V., Klypin
A. A., Primack J. R., Dekel A., 2001, MNRAS, 321, 559
Carroll S. M., Press W. H., Turner E. L., 1992, ARA&A, 30, 499
Cattaneo A., Dekel A., Devriendt J., Guiderdoni B., Blaizot J., 2006, MN-
RAS, submitted (astro-ph/0601295) (C05)
Chapman S. C., Blain A. W., Ivison R. J., Smail I. R., 2003, Nat, 422, 695
Chapman S. C., Smail I., Blain A. W., Ivison R. J., 2004, ApJ, 614, 671
Ciotti L., Pellegrini S., Renzini A., D'Ercole A., 1991, ApJ, 376, 380
Coil A. L. et al., 2004, ApJ, 609, 525
Courteau S., Rix H., 1999, ApJ, 513, 561
Dalcanton J. J., Yoachim P., Bernstein R. A., 2004, ApJ, 608, 189
De Lucia G., Kauffmann G., White S. D. M., 2004, MNRAS, 349, 1101
Dekel A., 1981, A&A, 101, 79
Dekel A., Silk J., 1986, ApJ, 303, 39
Dekel A., Woo J., 2003, MNRAS, 344, 1131
Dekel A., Arad I., Devor J., Birnboim Y., 2003a, ApJ, 588, 680
Dekel A., Devor J., Hetzroni G., 2003b, MNRAS, 341, 326
Dickinson M., Papovich C., Ferguson H. C., Budavari T., 2003, ApJ, 587,
25
Ding J., Charlton J. C., Bond N. A., Zonak S. G., Churchill C. W., 2003,
ApJ, 587, 551
Dressler A., 1980, ApJ, 236, 351
Dutton A., van den Bosch F. C., Courteau S., Dekel A., 2005, preprint (astro-
ph/0501256)
El-Zant A. A., Hoffman Y., Primack J., Combes F., Shlosman I., 2004a, ApJ,
607, L75
El-Zant A. A., Kim W., Kamionkowski M., 2004b, MNRAS, 354, 169
Fall S. M., Efstathiou G., 1980, MNRAS, 193, 189
Fall S. M., Rees M. J., 1985, ApJ, 298, 18
Fardal M. A., Katz N., Gardner J. P., Hernquist L., Weinberg D. H., Dave
R., 2001, ApJ, 562, 605
Ferguson H. C., Babul A., 1998, MNRAS, 296, 585
Field G. B., 1965, ApJ, 142, 531
Fioc M., Rocca-Volmerange B., 1999, A&A, 344, 393
Furlanetto S. R., Schaye J., Springel V., Hernquist L., 2003, ApJ, 599, L1
Giavalisco M. et al., 2004, ApJ, 600, L103
Gnedin N. Y., 2000, ApJ, 542, 535
Haiman Z., Rees M. J., Loeb A., 1996, ApJ, 467, 522
Hammer F., Flores H., Elbaz D., Zheng X. Z., Liang Y. C., Cesarsky C.,
2005, A&A, 430, 115
Hartwick F. D. A., 2004, ApJ, 603, 108
Hatton S., Devriendt J. E. G., Ninin S., Bouchet F. R., Guiderdoni B., Vibert
D., 2003, MNRAS, 343, 75
Heavens A., Panter B., Jimenez R., Dunlop J., 2004, Nat, 428, 625
Helsdon S. F., Ponman T. J., 2003, MNRAS, 340, 485
Hogg D. W. et al., 2003, ApJ, 585, L5
Kannappan S. J., 2004, ApJ, 611, L89
Kauffmann G. et al., 2003a, MNRAS, 346, 1055
Kauffmann G. et al., 2003b, MNRAS, 341, 54
Kauffmann G., White S. D. M., Heckman T. M., Menard B., Brinchmann J.,
Charlot S., Tremonti C., Brinkmann J., 2004, MNRAS, 353, 713
Keres D., Katz N., Weinberg D. H., Dave R., 2005, MNRAS, 363, 2
Klypin A., Zhao H., Somerville R. S., 2002, ApJ, 573, 597
Koushiappas S. M., Bullock J. S., Dekel A., 2004, MNRAS, 354, 292
Kravtsov A. V., 2003, ApJ, 590, L1
Kravtsov A. V., Berlind A. A., Wechsler R. H., Klypin A. A., Gottloeber S.,
Allgood B., Primack J. R., 2004, ApJ, 609, 35
Kravtsov A. V., Gnedin O. Y., 2005, ApJ, 623, 650
Kurk J., Rottgering H., Pentericci L., Miley G., Overzier R., 2003, New
Astron. Rev., 47, 339
Lahav O., Rees M. J., Lilje P. B., Primack J. R., 1991, MNRAS, 251, 128
Loeb A., Barkana R., 2001, ARA&A, 39, 19
Ma C., Boylan-Kolchin M., 2004, Phys. Rev. Lett., 93, 021301
Madau P., Ferguson H. C., Dickinson M. E., Giavalisco M., Steidel C. C.,
Fruchter A., 1996, MNRAS, 283, 1388
Madgwick D. S. et al., 2003a, ApJ, 599, 997
Madgwick D. S., Somerville R., Lahav O., Ellis R., 2003b, MNRAS, 343,
871
Maller A. H., Bullock J. S., 2004, MNRAS, 355, 694
Maller A. H., Dekel A., 2002, MNRAS, 335, 487
Maller A. H., Dekel A., Somerville R., 2002, MNRAS, 329, 423
Marinoni C., Hudson M. J., 2002, ApJ, 569, 101
Mathews W. G., Brighenti F., 2003, ARA&A, 41, 191
Mo H. J., White S. D. M., 2002, MNRAS, 336, 112
Mo H. J., Mao S., White S. D. M., 1998, MNRAS, 295, 319
Moustakas L. A. et al., 2004, ApJ, 600, L131
Murray N., Quataert E., Thompson T. A., 2005, ApJ, 618, 569
Nagai D., Kravtsov A. V., 2003, ApJ, 587, 514
Navarro J. F., Steinmetz M., 2000, ApJ, 538, 477
Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490, 493
Nilsson K., Fynbo J. P. U., Moller P., Sommer-Larsen J., Ledoux C., 2005,
A&A Lett., submitted (astro-ph/0512396)
Osmond J. P. F., Ponman T. J., 2004, MNRAS, 350, 1511
Ostriker E. C., 1999, ApJ, 513, 252
Pen U., 1999, ApJ, 510, L1
Prochaska J. X., Gawiser E., Wolfe A. M., Castro S., Djorgovski S. G., 2003,
ApJ, 595, L9
Rees M. J., Ostriker J. P., 1977, MNRAS, 179, 541
Ruszkowski M., Bruggen M., Begelman M. C., 2004, ApJ, 615, 675
Scannapieco E., Oh S. P., 2004, ApJ, 608, 62
Schaye J., Aguirre A., Kim T., Theuns T., Rauch M., Sargent W. L. W., 2003,
ApJ, 596, 768
Shapley A. E., Erb D. K., Pettini M., Steidel C. C., Adelberger K. L., 2004,
ApJ, 612, 108
Shaviv N. J., Dekel A., 2003, preprint (astro-ph/0305527)
Sheth R. K., Tormen G., 2002, MNRAS, 329, 61
Silk J., 1977, ApJ, 211, 638
Slyz A., Devriendt J., Bryan G., Silk J., 2005, MNRAS, 356, 737
Smail I., Ivison R. J., Blain A. W., Kneib J.-P., 2002, MNRAS, 331, 495
Sutherland R. S., Dopita M. A., 1993, ApJS, 88, 253
Thomas D., Maraston C., Bender R., de Oliveira C. M., 2005, ApJ, 621,
673
Thoul A. A., Weinberg D. H., 1995, ApJ, 442, 480
Tremaine S. et al., 2002, ApJ, 574, 740
Tremonti C. A. et al., 2004, ApJ, 613, 898
Voigt L. M., Fabian A. C., 2004, MNRAS, 347, 1130
White S. D. M., Frenk C. S., 1991, ApJ, 379, 52
White S. D. M., Rees M. J., 1978, MNRAS, 183, 341
Yan R., Madgwick D. S., White M., 2003, ApJ, 598, 848
Yang X., Mo H. J., van den Bosch F. C., 2003, MNRAS, 339, 1057
-3
v
v
2
v
0
This paper has been typeset from a TEX/LATEX file prepared by the author.