Astron. Astrophys. 363, 851-862 (2000)

5. Sensitivity of faint counts to the star formation history

Our simple SAM is not able to compute either the merging history of halos, or of the galaxies they host. However, we know that locally there is a tight correlation between major mergers on one side, LIRGs and ULIRGs on the other: at least 95 % of them are currently undergoing major mergers (see for instance Sanders & Mirabel 1996). It also seems fairly safe to assume that ISOPHOT and SCUBA sources are the high-redshift counterparts of such mergers. As a matter of fact, one could sum up the qualitative information from currently available datasets as follows. First, the objects seen by SCUBA have to be either very massive, or very efficient to extract energy from the gas, simply because their bolometric luminosity is larger than . Second, they have to be highly extinguished because most of this luminosity is emitted in the IR/submm. Third, for such numerous bright sources not to have been detected in the IRAS NEPR redshift survey at 60 µm, they have to be located in majority at redshifts greater than about , which seems to be the case for some of the SCUBA sources (Barger et al. 1999b).

In light of these observational facts, and as in GBHM, we define an ad-hoc "starburst" model, simply by pushing the limits of our quiescent models (SCDM, OCDM, or CDM), still powering the sources with star formation according to a Salpeter IMF. This consists merely in transforming a fraction of high-redshift quiescent objects into ULIRGs, while keeping all the parameters of the model fixed. The obvious interest of such an exercise is to assess whether one is able to reproduce the SCUBA source counts, along with preserving the quality of the fits of the optical counts used to calibrate the quiescent model, in the various cosmologies. We hereafter focus on the SCDM cosmology, for which the "quiescent" mode of star formation is unable to reproduce the submm counts.

In order to build such an ad-hoc model, we use the reasonable recipe that follows:

1. A fraction of objects with halo masses larger than goes through a ULIRG phase when their host halos collapse; their SFRs and optical depths are typically two orders of magnitude higher than those of the Milky Way (e.g. Rigopoulou et al. 1996). We tune our and parameters to obtain such properties for the heavily-extinguished burst mode of star formation. Typically, we take and for the starbursts.

2. These ULIRGs are mainly located at , which we enforce by requiring that their fraction evolves proportionally to the squared density, i.e. as .

3. Their number density at redshift 0 is consistent with the IRAS luminosity function of Soifer & Neugebauer (1991).

As a result of this phenomenological recipe, a typical halo of mass , with reduced spin parameter , that collapses at redshift , hosts by redshift (180 Myr after the starburst was triggered) a ULIRG of size 1 kpc that has consumed 98 % of its of cold gas initially present. The star formation rate averaged over this period is yr^-1. The starburst galaxy has a typical column density of about , and a metallicity of 0.03, yielding a face-on optical depth in the B band of 128. Its absolute B magnitude and bolometric IR luminosity (between 3 and 1000 µm) reach and respectively.

Of course, such a model is quite drastic, but once again, it should be considered as the necessary extension of the quiescent models to produce the correct amount of FIR/submm luminosity. The interesting result is that such a SCDM model in which all massive objects that form at redshifts higher than 1.5 are ULIRGs produces almost enough IR/submm luminosity to match the ISOPHOT and SCUBA counts, as can be seen in Fig. 4. This is also the typical luminosity one can extract from star formation with a Salpeter IMF without ruining the UV/IR calibration of the counts. For instance, decreasing the mass above which the ULIRG phenomenon occurs by an order of magnitude strongly decreases the optical counts.

Fig. 4. IR counts for the SCDM "quiescent" model (solid lines), the "burst" model (dots), and a still more efficient model to extract luminosity from the gas (dashes). The luminosities (resp. number densities) of ULIRGs are multiplied (resp. divided) by a factor 2 in the efficient model as compared to the "burst" model.

We also have to examine the possibility that a more efficient mechanism powers these sources, for instance a top-heavy IMF, with all the energy available through stellar nucleosynthesis being reprocessed in the IR/submm. The main features of such a model have been discussed in GHBM who take this solution to accommodate submm counts easily in an SCDM cosmology. We refer the reader to that paper for details. To test this possibility, we simply take our burst model and multiply the luminosity output of each ULIRG in the infrared per unit mass, , by a factor 2, while lowering the number of ULIRGs in the model by 2. This is to say, we trade the number of sources for more luminosity per source. Fig. 4 and Fig. 5 show that the influence of such a redistribution on the counts is weak. Of course, any combination of luminosity and number density of ULIRGs is possible.

Fig. 5. Multi-wavelength redshift distributions for the SCDM "quiescent" model (solid lines), the "burst" model (dots), and a still more efficient model to extract luminosity from the gas (dashes), as in Fig. 4.

In light of the previous work, and bearing in mind that we want to describe multi-wavelength galaxy counts, we can define a "best guess" model within a given cosmological model. We hereafter retain the CDM model as a typical example, since the optical and submm counts with the CDM model and the quiescent mode of star formation only are intermediate between the SCDM and OCDM. We take , , , , and . In terms of our astrophysical parameters, we keep the standard value , and we take , for the quiescent galaxies, and , for the starbursts. One ULIRG dwells in each halo that is more massive than and collapses before redshift 1.5. This ULIRG population evolves as the density squared at lower z, so that at redshift 0, its number density is about Mpc^-3, corresponding to only one ULIRG for 2500 halos . The predictions for the faint counts are given in Fig. 6 and Fig. 7. The model provides a good fit of the faint counts at optical wavelengths (though the bright counts are slightly overestimated). The quality of the fit nicely compares with other faint counts obtained from SAMs (e.g Kauffmann et al. 1994). The ISOCAM 15 µm data and IRAS 60 µm data are also fairly reproduced, though the observed slope of the 15 µm counts seem to be slightly steeper than the model. The fit of the submm counts is also very satisfactory.

Fig. 6. UV/near-IR counts for the fiducial model (solid line).

Fig. 7. Mid-IR/submm counts for the fiducial model (solid line).

The redshift distributions are given in Fig. 8. The CFRS predictions now peak almost at the correct redshift. The NEPR predictions still exhibit a high-redshift tail as in GHBM, in contrast with the data, but the level is much lower than in GHBM. We recall that the NEPR sample is polluted by a supercluster in the first redshift bin. Moreover, a recent follow-up of this sample with ISOCAM at 15 µm seems to show that some of the sources are multiple and that the optical identifications might be ambiguous in these cases (Aussel et al. 2000). The relative levels of the two peaks in the redshift distribution at 175 µm are sensitive to the flux cut-off. Most of the sources in the redshift distribution for the SCUBA deep surveys at 850 µm are predicted to be at , but the comparison with data is still difficult because of identification uncertainties (see e.g. Barger et al. 1999b corrected after Smail et al. 1999).

Fig. 8. Multi-wavelength redshift distributions for the fiducial model (solid line).

Finally, Fig. 9 shows the Cosmic Background obtained by integrating the faint counts, and compare the predictions with current data in the optical, IR and submm. Whereas introducing ULIRGs in an ad-hoc way into our simple models suffices to reproduce the Cosmic IR Background and the submm counts at 850 µm, it falls marginally short of getting the required diffuse background flux at 140 and 240 µm, though it reproduces the ISOPHOT counts brighter than 100 mJy at 175 µm. These galaxies contribute only 10 % of the background. So this discrepancy may be due only to the fact that the 175 µm counts below 100 mJy are much steeper than our predictions. The model is too low by a factor of 2 with respect to the points corrected for warm galactic dust by Lagache et al. (1999), which are themselves a factor of 1.5 below the points without such a correction by Hauser et al. (1998). The difficulty to fit the points might indicate that this correction is still underestimated. Finally, one should also be aware that a contribution of intergalactic dust (with a grey extinction curve) to the background light is also possible (Aguirre & Haiman 2000). Adding these extra components might help reconcile models and observations.

Fig. 9. Diffuse Background light for the fiducial model (solid line).

We conclude from these figures that this fiducial model gives a satisfactory estimate of the luminosity budget of galaxies, and allows us to interpolate or extrapolate the observed faint counts to other wavelengths and fainter flux levels.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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