The annual birthrate is proportional to the SFR in the galaxy where is the timescale for star formation. The SFR adopted is a decreasing exponential, which appears when the SFR is supposed to be proportional to the gas content without taking into account the gas ejected by stars (e.g. Bruzual & Charlot 1993):
where the age of the galaxy is given by and the time at which the galaxy was formed. At any time t the merger rate has a contribution from systems that are formed at different epochs in the history of that galaxy. It can thus be expressed as a convolution integral of the birthrate of stars and the distribution of merging times :
The results of the population synthesis computations give and the normalization coefficient . The function from PZY98 is approximated by a Gaussian with as a parameter, the maximum at and with =1 ( is the age of the galaxy at ). We assume that the distribution function , normalized to unity, is time-independent and the same for all galaxies. For the normalization of we require that in a reference galaxy similar to the Milky Way (type Sb, , with a current astration rate of ) the current rate of mergers is yr-1, which is the merger rate obtained for the Galaxy by PZY98.
For simplicity, we split the Hubble sequence into three types of galaxies: E to SO, Sa to Sb and Sc to Sd. For our selected mixture of Hubble types, two sets of characteristic star formation timescales are used (cf. Table 1). This parameterization is rather simplistic; the fraction of E/S0 galaxies may differ considerably from 20% (Dressler 1980) and the star formation history in spirals of the same morphological type may be a function of their mass (Gallagher et al. 1984). If indeed most stars in the Universe formed in dwarf star-burst galaxies (e. g. Babul & Ferguson 1996) at , this may affect our results considerably.
Table 1. Adopted parameters for galaxies of different morphological types: timescale of star formation ; contribution to the B -band luminosity of the Universe (Phinney 1991); mass to blue-light ratio (after Lipunov et al. 1995 and Guiderdoni & Rocca-Volmerange 1987).
Star formation is assumed to occur continuously according to Eq. 1in all galaxies. We also investigate the case of an initial burst of star formation during the first Gyr in E-SO galaxies and no star formation thereafter. The latter models are denoted as sets 1b and 2b.
The comoving rate density can be related to via the B-band luminosity density from the Universe (Efstathiou et al. 1988) where :
Here the summation is taken over all morphological types and is the contribution of each type of galaxy to the B-band luminosity of the Universe, is in solar units (see Table 1).
The comoving rate density depends on the cosmological parameters and on the redshift of formation , via the relation for the age of galaxies relative to redshift: . We consider two cosmological models with and no vacuum energy: and (0.75, 0.2) for which =12.16 and 9.92 Gyr, respectively.
© European Southern Observatory (ESO) 1998
Online publication: March 30, 1998