Morphological features of ULIGs suggest that most ULIGs come from
mergers between or among galaxies (Sanders et al. 1988a; Taniguchi, & Shioya 1998).
Another important property of the ULIGs is that they are very gas rich;
e.g., 
(Sanders et al. 1988a; Scoville et al.
1991; Downes & Solomon 1998). Therefore, the progenitor galaxies of ULIGs are
rich in gas such as giant spiral galaxies. Since intense starbursts are observed
in many ULIGs, the most probable formation mechanism of SMBHs is the collapse
of compact remnants of massive stars (Weedman 1983; Norman & Scoville 1988).
Another important issue is whether or not progenitor
galaxies had SMBHs originally in their nuclei.
Although masses of SMBHs in nearly spiral galaxies (i.e., progenitor galaxies
of ULIGs) are of the order of
at most (e.g., Kormendy et al. 1998 and references therein),
these seed SMBHs could grow due to gas accretion in circumnuclear dense-gas
regions during the course of the merger. Therefore, we consider two cases;
1) at least one progenitor had a SMBH with 
,
and 2) no progenitor had a SMBH.
If a progenitor galaxy had a SMBH in its nucleus, this seed SMBH could grow
in mass during the course of merger because the central region of a ULIG
is very gas-rich.
Such gas accretion may be efficient within the central 1 kpc region because
the gas density is quite high in the central region; i.e., 
cm
(Scoville, Yun, & Bryant 1997; Downes & Solomon 1998).
We here consider the classical Bondi-type (Bondi 1952) gas accretion
onto the SMBH. This gas accretion rate is given by
where 
, 
, 
, and 
are the mass of
a hydrogen atom, the number density of the hydrogen atom,
the accretion radius defined as 
(
is the mass of the seed SMBH), and the effective relative velocity
between the seed SMBH and the ambient gas, respectively.
A typical dynamical mass of the central 1 kpc region is of the order of
. Suppose that the SMBH with mass of 
is sinking toward the dynamical center of the merger. Its orbital velocity is
km s
,
where 
is the dynamical mass of the central 1 kpc region
of the merger and 
is the radius in units of 1 kpc.
The crossing time of the SMBH is 
years.
Therefore, the merging
time scale is estimated to be 
years (e.g., Barnes 1989).
Adopting an average gas density in the circumnuclear regions of ULIGs
to be 
cm
,
we obtain the accreting gas mass during the course of merger;
where 
is the mass of SMBH in units of 
,
is the average gas density in units of 
cm
,
is the orbital velocity with respect to the ambient gas
in units of 100 km s
, and 
is the merger
time scale in units of 
years.
This estimate implies that the seed SMBH cannot grow up
to 
if the seed SMBH is less massive than
. Since the ULIGs come from
mergers between two galaxies or among several galaxies,
their progenitor galaxies should be very giant spiral galaxies
in order to pile up molecular gas up to 
in their
central regions (e.g., Sanders et al. 1988a).
In fact, nearby spiral galaxies such as M31
and NGC 4258 have SMBHs with a few 
[e.g., Kormendy et al.
(1998) and references therein; see also Miyoshi et al. (1995)
for the case of NGC 4258]. Therefore, it seems quite likely that
the seed SMBH may be more massive than that adopted in the above estimate.
If the seed SMBH is more massive than a few
, it could grow up to 
.
Although we have no knowledge about the seed SMBHs in the progenitors,
our estimates given here suggest that the gas accretion in the dense gas
clouds onto the seed SMBH can lead to the formation of a quasar nucleus
in the heart of ULIGs.
Next we consider a case where there is no seed SMBH in the progenitor
galaxies of ULIGs. In this case, a possible way to form quasar nuclei in ULIGs
is to pile up the circumnuclear star clusters of compact remnants of massive stars;
black holes and neutron stars, each of which has a few 
at most.
This issue was pioneeringly discussed by Weedman (1983).
Using both starburst models by Gehrz, Sramek, & Weedman (1983) and optical
spectroscopic observations (i.e., H
luminosity), he suggested that
starburst galaxies with L(H
) 
erg s
could
produce compact starburst remnants up to 
.
However, since the H
luminosity of starburst galaxies is dominated by
the most massive stars in the starburst region, it seems hard to
estimate the total mass of compact remnants solely using L(H
).
Thus, the estimate by Weedman (1983) may provide a rough upper limit for the
compact remnant mass in the starburst galaxies.
Norman & Scoville (1988) also discussed the formation of quasar nuclei
in ULIGs. However, their assumption (a coeval, massive-star cluster of
within the central 10 pc region) turns out to be
unlikely because the recent high-resolution optical and near-infrared imaging
by the Hubble Space Telescope of the ULIGs have shown that blue star clusters
are distributed in the circumnuclear regions up to 
a few kpc
(Shaya et al. 1994; Surace et al. 1998; Scoville et al. 1998).
Although the central star cluster associated with the western nucleus of
Arp 220 is very luminous and its mass is estimated to be 
,
typical masses of the circumnuclear star clusters are of the order of
at most
(Shaya et al. 1994; Scoville et al. 1998; Shioya, Taniguchi, & Trentham 1998;
see also Taniguchi, Trentham, & Shioya 1998).
Although some star clusters may be hidden because of heavy extinction
(Scoville et al. 1991; Genzel et al. 1998),
we firstly investigate whether or not the star clusters
found both in the optical and in the near infrared (NIR) (Shaya et al. 1994; Scoville
et al. 1998) will be responsible for the formation of a SMBH with
.
Shaya et al. (1994) discussed the fate of
circumnuclear star clusters; since these clusters will lose their kinetic energy
to individual stars during random encounters (i.e., dynamical friction),
they will sink toward the merger center within 
years.
Here, from a viewpoint of dynamical relaxation of the star clusters,
we examine whether or not compact remnants formed in
the circumnuclear star-forming clusters can make a SMBH with
.
For simplicity, we consider a case where ten circumnuclear star-forming
clusters, each of which has a total stellar mass of 
, are distributed
within 
a few kpc. It is known that stars with 
produce compact remnants. We estimate how many such massive stars are
formed in each cluster.
We assume that stars are formed with a Salpeter-like initial mass function (IMF);
where m is the stellar mass in units of 
and
is a normalization constant determined by the relation
which leads to
The number of stars with a mass range 
is estimated as
Using 
in equation (5), we re-write equation (6) as
There are three free parameters; the power index (
), and the upper and
lower mass limits of the IMF (
and 
). Since stars with 
produce compact remnants, we set 
and 
.
In Table 1, we give results for some possible combinations of the parameters.
Although 
= 1.35 is the canonical value for stars in the solar neighborhood,
there are some lines of evidence that massive stars
are more preferentially formed in such violent star-forming regions;
i.e., a top-heavy initial mass function with 
(e.g., Goldader et al. 1997 and references therein).
Therefore, we adopt a case of 
, 
and
. In this case, there are 
massive stars
with 
in each cluster. Each compact remnant has a mass from
a few 
(for neutron stars) to several 
(for black holes).
Therefore, the total mass of compact remnants in each cluster is
.
Such ten clusters will be relaxed dynamically with a time scale of
where 
is the typical size of a circumnuclear region with
ten star clusters in units of 1 kpc and
is the mass of the clusters in units of 
.
Therefore, we expect that a SMBH with mass of 
will be made
years after the onset of circumnuclear starbursts in ULIGs.
This mass is smaller than that necessary for quasar nuclei
(i.e., 
). Each cluster of compact remnants would be able
to gain its mass by gas accretion as discussed in section 2.1. However,
the accreted mass is estimated to be 
for
ten clusters in total, being still less massive than 
.
Here it should be again remembered that all star clusters in the central region of
ULIGs cannot be observed in both the optical and the NIR because
inferred extinction toward the nuclei of ULIGs is very large; e.g.,
50 mag (Genzel et al. 1998; Scoville et al. 1991). Therefore,
it is quite likely that the majority of nuclear star clusters in ULIGs
are hidden by a large amount of gas and dust.
Recently, Shioya et al. (1998) analyzed the optical-NIR spectral energy
distributions of nuclear star clusters in Arp 220. They found that
these clusters are more massive systematically (i.e., 
)
than circumnuclear ones but can account only for about one-seventh of
the total bolometric luminosity of Arp 220. Although OH megamaser sources found
in the central region of Arp 220 may provide possible evidence for hidden AGN
(Diamond et al. 1989; Lonsdale et al. 1998), the recent mid-infrared spectroscopy
of a sample of ULIGs has shown that the majority of the ULIGs such as Arp 220
are powered by nuclear starbursts (Genzel et al. 1998; Lutz et al. 1998).
Therefore, it is strongly suggested that some hidden star clusters should be
responsible for the remaining (
) bolometric luminosity of Arp 220.
Since compact remnants produced in the hidden clusters will join to form a SMBH,
it is expected that a SMBH with 
will be formed
in the heart of ULIGs.