Within the galaxy formation paradigm of White & Rees (1978) and Fall & Efstathiou (1980), the DM haloes acquire their spin from tidal interactions and baryons increase their rotational support by falling into the potential wells of these haloes and by conserving their angular momentum. To what degree the baryon angular momentum is preserved during this process has emerged as one of the key issues in our understanding of disk galaxy formation.
2.1. The angular momentum catastrophe
So is there a problem with J when forming galactic disks? We
start with the so-called angular momentum `catastrophe' which has
emerged from the comparison of observations with numerical simulations
of disk formation. While the observed disks have shown a deficiency of
specific angular momentum j by a factor of two, modelled disks
appear to have radial scalelengths smaller by a factor of 10 compared
with observations, resembling bulges rather than disks (e.g.,
Navarro & Steinmetz
2000).
In other words, the actual disks rotate too fast
for a given luminosity L. At the same time, the slopes of the
modelled disks turned out to be in agreement with the I-band
Tully-Fisher (TF) relation for late-type galaxies, L ~
vmax, where
~ 2.5-4,
increasing to ~ 3 in I. In our context, this is important because
TF defines the relation between the dynamical mass and L.
Simulations have shown much more compact endproducts of baryonic
collapse, where the gas has lost a substantial fraction of its
j. As we shall see below, the possible remedy to this problem
involves a combination of numerical and physical factors, but no
universal solution has been found so far.
What are the possible contributors to the angular momentum catastrophe? The first suspect is lack of numerical resolution. Indeed, increasing the resolution by adding smooth particle hydrodynamics (SPH) particles (e.g., Lucy 1977; Monaghan 1982) has improved the situation by helping to resolve the maxima of rotation curves, resulting in flatter and lower curves, thus having an effect on the central and outer regions of modelled galaxies (e.g., Governato et al. 2004; Naab et al. 2007; Mayer et al. 2008).
Numerical non-conservation of j is another suspect. In the outer disks, it can cause an excess of angular momentum transfer to the halo (e.g., Okamoto et al. 2005). In the inner disks, this effect could smear the radial gradients of rotation velocity, and artificially stabilise the disks against bar instability. Modelling an isolated galaxy formed from a non-cosmological Milky Way-type DM halo has demonstrated that low-resolution disks heat up and lose j to the halo. Colder disks remain larger and have smaller bulge-to-disk ratios. A possible solution lies in the increase of the number of SPH particles and the introduction of a multi-phase interstellar medium (ISM).
Next, the disk ISM cooling below the
Toomre (1964)
threshold of
Q
cs
/
G
= 1 leads to an
excessive fragmentation (we return to this problem when discussing
over-cooling). Here cs is the sound speed in the gas,
- the epicyclic
frequency, and
-
the disk (gas) surface density. The multi-phase ISM introduced by
Springel &
Hernquist (2003)
through a modified equation of state clearly alleviated the problem. The
gas cooling must be compensated by some kind
of feedback to avoid fragmentation and the follow-up runaway star formation.
This feedback from stellar evolution, e.g., from SNe and OB stars,
has a clear effect on the bulge-to-disk ratio
(Robertson et al.
2004;
Heller et al.
2007b).
Finally, it was pointed out that baryons can hitchhike inside DM halo substructures which spiral in to the centre as a result of dynamical friction (e.g., Maller & Dekel 2002). This process is similar to that in a disk, where in the absence of star formation, dense gas clumps would heat up the disk, spiral in to the centre and possibly contribute to the bulge growth (e.g., Shlosman & Noguchi 1993; Noguchi 1999; Bournaud et al. 2007). This process, related to overcooling, can be easily over- or underestimated because it depends on a number of additional processes, e.g., gas-to-stars conversion, ablation, etc. (see Section 6.1). Also, one cannot neglect the contribution of clumpy DM and baryons to the mechanical feedback (i.e., by dynamical friction), and, consequently, to the fate of the central DM cusp and the formation of a flat density core in the halo (e.g., El-Zant et al. 2001; Tonini et al. 2006; Romano-Díaz et al. 2008a).
2.2. The cosmological spin distribution and individual haloes
We now turn to the distribution of angular momentum in DM haloes. Early considerations of the baryon collapse assumed an initial solid-body rotation for uniform-density gas spheres. The angular momentum distribution (AMD) can be written as M(< j) = Mh[1 - (1 - j / jmax)3/2], where jmax is some maximum j, if baryons preserve j during this process, which results in exponential disks (e.g., Mestel 1963). More generally, what is the shape of an AMD which leads to exponential disks? The initial solid-body rotation law is soon replaced, following assumptions that baryons are mixed well with the DM and share its j. Assuming j conservation, Md(< j) / Md = Mh(< j) / Mh, where Md is the disk mass (e.g., Fall & Efstathiou 1980; Bullock et al. 2001; Maller & Dekel 2002), the new AMD can be written as
![]() |
(3) |
and P(j) = 0 if j < 0. This leads to a mass distribution with j
![]() |
(4) |
with two parameters, µ - the shape, and j0 = (µ - 1) jmax - whose effect on M(< j) has been displayed in Bullock et al. (2001, Fig. 4) for four haloes. When using the best-fit j0, and normalising by Mh, all haloes follow the same curve nicely - an indication of the universality of the AMD. The resulting (computed) disk surface density is nearly exponential, but with an over-dense core and an overextended tail.
Using
Swaters' (1999)
sample of 14 dwarf disk galaxies,
van den Bosch et
al. (2001)
have demonstrated that the specific AMDs (i.e., M(< j))
of these disks differ from halo AMDs proposed by
Bullock et al.
(2001).
But the total disk j's, i.e.,
P(), are similar to
those of haloes. Although the pre-collapse baryon and DM AMDs are
expected to be similar, the disks formed have been found to lack low and
high j - in direct contradiction to the simple theoretical
expectations discussed above. Moreover, the mass fraction of Swaters'
disks appears to be significantly smaller than the universal baryon
fraction in a
CDM
universe. Does this mean that disks form from
a small fraction of baryons and, in spite of this, attract most of the
available angular momentum? One can argue that a redistribution of
baryon j should occur (at a stage earlier than anticipated), or
that the high-j baryons avoid the disks. What processes are
responsible for modifying the baryon AMDs? As argued by
van den Bosch et
al. (2001),
a plausible explanation can lie in the selective loss of the
low-j gas, leading to essentially bulgeless disks.
The closely-related issue of the existence of bulgeless disk galaxies is discussed in Section 6.
2.3. The tidal torque theory and the origin of the universal angular momentum distribution
Rotation of galaxies results from gravitational torques near the time of the maximum expansion (e.g., Hoyle 1949; Doroshkevich 1970; White 1978). If J is calculated using the spherical shell approximation (i.e., linear tidal torque theory, hereafter TTT) further redistribution during the non-linear stage contributes little (Porciani et al. 2002a, b). Alternatively, J can be acquired in the subsequent non-linear stage of galaxy evolution via mergers, accounting for the orbital angular momentum and its redistribution (e.g., Barnes & Efstathiou 1987).
According to the TTT, J is gained mostly in the linear regime of growing density perturbations due to the tidal torques from neighbouring galaxies. The maximum contribution to J comes from the times of the maximum cosmological expansion of the shell. Little J is exchanged in the non-linear regime after the decoupling and virialisation of the DM. Baryons are well mixed with the DM and acquire similar j (see Section 1). The TTT follows the evolution of the angular momentum of the DM halo based on the moment of inertia tensor and the tidal tensor. Within this framework, galactic spins can be estimated using the quasi-linear theory of gravitational instability. The TTT has been extensively tested by numerical simulations.
So far we have not considered the effect of the mergers on the evolution
of J (or ). We
define mergers as major if their mass ratio is larger than 1:3.
Wechsler (2001)
has argued that major mergers dominate the j-balance in
galaxies. Do they? The crucial point here is to account for the
redistribution of the angular momentum during and after the merger.
Hetznecker &
Burkert (2006)
have separated the main contributors to
~
J|E|1/2
M-5/2, i.e., J, E and
Mh, and followed them during and after
the merger event. While the DM halo J experiences a jump during a
major merger, the subsequent mass and energy redistribution in the halo
washes out the gain in J. A similar conclusion has been reached by
Romano-Díaz et
al. (2007)
and is displayed in Fig. 1. The corollary of
high-resolution DM simulations is that there is no steady increase in
the halo cosmological spin
with time due to
major mergers.
![]() |
Figure 1. Evolution of DM halo spin
|
To summarise, hydrodynamical simulations show that the angular momentum of baryons is not conserved during their collapse. A number of factors can contribute to this process, but there is no definitive answer yet. The proposed solutions often amount to fine tuning. There is also a mismatch in the j-profiles. The distribution of the modelled specific angular momentum does not agree with the observations.