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Astron. Astrophys. 350, 349-367 (1999)

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8. Discussion

8.1. Uncertainties in the models

A major source of uncertainty is the treatment of dust extinction. In Sect. 4 we showed how different assumptions about the dust extinction affect the estimated rates. An underlying assumption in each of the calculations is that the same amount of dust affects the UV-luminosity from the high-z galaxies (used to calculate the SNR) and the light from the SNe. If the UV-luminosity originates from regions with dust properties that differ considerably from the regions where the SNe originates, an extra dispersion in the estimated rates should be expected. It is, however, difficult to estimate the uncertainties in the rates introduced by this, since the distribution of the dust within the galaxies at high z is poorly known.

Other sources of uncertainties are the range of progenitor masses in Eq. (2) and the choice of IMF (Sect. 2.2), as well as the dependence on cosmology (Sect. 5). Also, the distribution of the different types among the core collapse SNe influences the estimates. Changes in the fractions of the faint SN1987A-like or bright Type IIn's are most important. Due to their low luminosity, the SN1987A-like SNe will be too faint to be detected by ground based telescopes for [FORMULA] 0.5 (IAB [FORMULA] 27 at peak). Doubling the fraction of these SNe from 15% to 30%, or, at the other extreme, setting the fraction to zero, changes the total number of detected SNe by [FORMULA]18% at these redshifts. Changing the fraction of Type IIn's has a slightly different effect. By increasing f(IIn) from 2% to 4%, which is the upper limit proposed by Cappellaro et al. (1997), the total number of detected core collapse SNe at IAB [FORMULA] 24 increases by [FORMULA] 17%. At these magnitudes it is mainly low redshift SNe in the steep part of the SFR curve that contribute, explaining the relative large effect. At fainter limiting magnitudes, an increasing fraction of normal luminosity SNe are detected at, and even beyond, the peak. The relative increase due to the Type IIn's is therefore marginal. For IAB = 27 the number of SNe increases by [FORMULA] 4% and for [FORMULA] the number increases by [FORMULA] 1%.

Adding the uncertainties, we find that the counts of core collapse SNe may vary by a factor more than two due to insufficiently known model parameters.

The estimated rates of Type Ia SNe are subjected to even larger uncertainties. Besides the factors which also affect the core collapse SNe, the rates of Type Ia's depend strongly on the assumed progenitor scenario, and are more dependent on cosmology. Also, the normalization at z = 0 introduces an additional uncertainty by a factor [FORMULA] 3. Considering this, it seems unlikely that counts of Type Ia's can be used to set any constraints on the model parameters. With additional information about the redshift distribution of the SNe it should, however, be possible to constrain either the progenitor life time or the cosmology. One of these parameters must, however, be independently determined, since there is a degeneracy between the both when it comes to the estimated redshift distribution of the SNe. To decrease the uncertainties involved, well defined searches of SNe at low redshift are highly desirable.

8.2. Dependence on metallicity

With increasing redshift the mean metallicity decreases, although the dispersion may be higher that at present. It is therefore interesting to discuss the consequences of a lower general metallicity.

For Type Ia SNe, Kobayashi et al. (1998) argue that for one of the most likely progenitor scenarios, based on super-soft X-ray binaries, the necessary condition for a Chandrasekhar mass explosion may not occur for a metallicity of [FORMULA] 0.1 [FORMULA], consistent with the decrease in the galactic [Fe/O] ratio at this metallicity. The physical reason can be traced to the peak in the opacity curve at [FORMULA] 105 K, produced by iron. In the scenario by Kobayashi et al., this corresponds to a drop in the Type Ia rate at [FORMULA] 1.2. At higher redshifts Type Ia explosions are inhibited, since [Fe/H] [FORMULA]1 here. Including the possibility of a dispersion in the evolution of the metallicity leads Kobayashi et al. to conclude that a cutoff in Type Ia rates should occur at z = 1-2. The Kobayashi et al. scenario should observationally be similar to the [FORMULA] = 3 Gyr model, with a turn-on at [FORMULA] 1.5. Because the life time in their model is [FORMULA] = 0.6 Gyr, the peak in the SNR should occur at [FORMULA] 1, rather than at [FORMULA] 0.7, as in the [FORMULA] = 3 Gyr model.

For core collapse SNe a lower metallicity can have several effects. First, line blanketing in the UV may decrease somewhat for the Type IIP's. Since this is mainly an ionization effect, rather than an abundance effect, it is, however, likely that this effect is small. This is partly confirmed by the calculations by Eastman et al. (1994), who find only a marginal decrease of the UV blanketing as the metallicity is decreased from solar to a tenth of solar. For extremely low metallicity, like for Pop. III stars with [FORMULA] [FORMULA], blanketing may, however, decrease significantly, although the Balmer jump will still be present.

Secondly, the relative fraction between different SN types may change. In particular, the number of blue, compact supergiant progenitors similar to Sanduleak [FORMULA] 202 may increase. This would then lead to a larger fraction of faint SN1987A like SNe, decreasing the number of observable core collapse SNe at high z. We emphasize that the exact reason for the blue progenitor of SN1987A is not fully understood (e.g., Woosley et al. 1999).

Finally, the mass loss process of the SN progenitor may depend on the metallicity. An example is the known decrease of the mass loss rate with decreasing metallicity for radiatively accelerated winds (e.g., Kudritzki et al. 1987). In the red supergiant phase dust driven mass loss may be less efficient (Salasnich et al. 1999). The importance of binary mass transfer may also depend on the metallicity. A change in the mass loss rate with metallicity would change the relative proportions between the different core collapse types. In particular, a decrease of the total mass lost is expected to lead to a decrease in the number of Type IIL, Ib and Ic SNe, while favoring Type IIP's. In addition, a less dense circumstellar medium medium could then lead to a decrease in the ionization by the circumstellar interaction, and stronger line blanketing for the Type IIL and IIn's. Note, however, that the mass loss process even for local red supergiants in their final phase is poorly understood.

8.3. SNe as probes of star formation

With large ground-based telescopes, and especially with NGST, it should be possible to detect SNe up to high redshifts, and to estimate the rates of both core collapse and Type Ia SNe. We have shown that because the rate of core collapse SNe follows the SFR, it should be possible to use observed rates of these SNe to constrain the SFR. As we have seen, a major problem is the influence of dust extinction. In this respect we note that the NIR bands have the advantage of being less affected by dust extinction than the observed UV-luminosities. At high redshifts these bands correspond to the optical rest wavelength bands, and have therefore a factor of [FORMULA] lower extinction than the UV bands. Different star formation models may therefore better be tested by using the K and M bands. Reaching high redshifts ([FORMULA] 2), corresponding to [FORMULA], increases the differences in the predicted counts between the star formation models significantly (see Sect. 4.3).

The estimated difference in the redshift-integrated counts between the hierarchical star formation model and the high dust star formation model is a factor [FORMULA] 2. This is about the same factor produced by the uncertainties in the modeling. It is therefore difficult to use these counts to probe star formation scenarios, unless the parameters involved are better known. What seems more feasible is to use redshift subsamples to constrain the shape of the SFR. As shown in Sect. 4.3, for models with a flat SFR at high z, we estimate a factor [FORMULA] [FORMULA] more SNe with [FORMULA] 4. A major problem here is to determine the redshifts of the SNe (see discussion in Sect. 7).

A further problem is if a large fraction of the star formation takes place in galaxies with very large extinction, like M 82 or Abell 220 with [FORMULA] magnitudes (instead of [FORMULA], as for our high dust model). The difference between the optical and UV extinction is then of less importance, leading to a decrease in the estimated differences between the models. A similar large extinction may be indicated by the results of the ISO observations of some deep fields (e.g., Flores et al. 1999). Far-IR observations is then the most reliable way of deriving the true star formation rate. An alternative is to use some other source tightly coupled to star formation, but not affected by dust absorption. If gamma-ray bursts are related to some class of core collapse SNe (e.g., Type Ic's), they may be such class of objects (Cen 1998). Also radio observations may be interesting in this respect.

8.4. SNe and nucleosynthesis

The study of the nucleosynthesis by direct observation of SNe is naturally affected by the same problem as the star formation rate. Unless the dust extinction can be determined reliably in an independent manner, the true number of SNe is difficult to derive. In addition to this, the metallicity yields for SNe of different masses is non-trivial to derive even at low redshifts (e.g., Fransson & Kozma 1999). Only for SN 1987A and a couple of other SNe has this become possible. The alternative to use theoretical yields from collapse calculations is obviously less satisfactory. The lower metallicity may also affect e.g. the mass loss processes, as discussed in Sect. 8.2. This may change both the progenitor structure and the upper and lower limits for the core collapse and Type Ia SNe, as well as the heavy element yields. In our view one of the most interesting goals for the observation of SNe at high redshift may be to observationally study the differences between the SNe in the early universe and those today.

8.5. Spacing between observations

An important aspect concerning the detection of SNe is the spacing in time between the observations. In order to detect the SN, the magnitude has to change appreciably. The interval is primarily dependent on the shape of the light curve. Near the peak, where the SN changes relatively fast, a comparatively short time is sufficient. This applies to searches where detection of SNe on the rising part of the light curve is the main objective (e.g., searches for Type Ia's for [FORMULA]). Core collapse SNe, which have a flatter decline of the light curve, need a longer spacing. This is especially true for the Type IIP's, which in the plateau phase decline by [FORMULA] 1 mag. Unless a SN can be detected (against the host galaxy) with this precision it will be missed. The limiting magnitude of the search also affects the necessary spacing. A deeper search results in a higher mean z of the observed SNe. Due to the cosmic time dilatation, the light curves of these SNe are stretched in time, implying that a longer interval between two observations is needed, [FORMULA] 100(1 + z) days. Therefore, in order to detect these SNe a deep observation with the VLT requires an interval of [FORMULA] 100 days, while a corresponding NGST observation requires approximately a years interval.

8.6. Comparison to other works

Apart from the studies by Pain et al. (1996), Madau et al. (1998a) and Chugai et al. (1999), which we have already commented on, there is a number of related investigations.

Marri & Ferrara (1998) have studied of the effects of gravitational lensing of high redshift SNe. Using a Press-Schechter formalism and gravitational ray-tracing, they determine the magnification probability as function of redshift for different cosmologies. We have already discussed the implications of their lensing results for our simulations in Sect. 6. Marri & Ferrara use these magnification probabilities to estimate the observed magnitudes at high redshift. The fact that there is a relatively large probability, [FORMULA] 10% for a factor of three or larger magnification for [FORMULA] 4, means that even SNe as distant as [FORMULA] 10 may be within the limits of NGST. When estimating the observed magnitudes they, however, assume that the light curve is described by a Type IIP light curve without any dispersion in magnitude, although as we have seen, the Type IIP's show a very large variation in luminosity. They also assume a fairly high temperature, [FORMULA] 25 000 K during the first 15 days, which is twice as high as the models by Eastman et al. (1994) give. This is especially important for the high-z SNe, and, as Marri & Ferrara show, a lower temperature makes the detectability considerably more difficult. Marri & Ferrara do not attempt any discussion of expected rates of the high-z SNe.

The effects of gravitational lensing is also investigated by Porciani & Madau (1998). They find, as earlier mentioned, a considerably lower probability for a substantial magnification than Marri & Ferrara do. Porciani & Madau present I band counts for Type Ia and core collapse SNe, both including GL, and without lensing. These counts are presented as the number of SNe in different magnitude bins (21 [FORMULA] IAB [FORMULA] 27), seen at the peak of the light curve for an effective observation duration of one year. This leads to lower estimates for the observable number of SNe compared to our estimates, where we include SNe detected over the whole light curve. To compare our results we have calculated counts in the same units as used by Porciani & Madau. We find a fairly good agreement between the core collapse counts (deviation by a factor [FORMULA] 2), but a somewhat worse agreement between the Type Ia counts. It should be noted that expressing rates in units of an effective observation duration requires an idealized observational procedure (as we have shown in the examples in Sect. 3.1).

Ruiz-Lapuente & Canal (1998) discuss the possibility of using R band counts of Type Ia SNe to distinguish different progenitor scenarios. They find, similar to our estimates, that models with long-lived progenitors result in higher counts than models with short-lived progenitors. To use this as a probe they note that it is necessary to know the SFR better than a factor 1.5. However, the uncertainty in the SFR seems, as we have shown, to be larger than this. It should therefore be difficult to use counts to determine progenitor scenarios. Additional information about the redshift distribution of the SNe is required.

The same authors also estimate the effects on the counts for alternative cosmologies. They find that a flat [FORMULA]-dominated universe ([FORMULA]CDM) should result in higher counts of Type Ia SNe than a standard cold dark matter universe (SCDM). The difference between the cosmologies start at mR [FORMULA] 24, and increases at fainter limiting magnitudes. A somewhat smaller increase in the counts is found for an open universe with zero cosmological constant (OCDM).

Our results for different cosmologies agree with the general trend of Ruiz-Lapuente & Canal. Using counts to distinguish between cosmologies, however, requires both that the SFR is well known, and that restrictions can be set on the progenitor life time. If this is not the case, the degeneracy between the different parameters involved makes a distinction between cosmologies very difficult.

In an interesting paper Miralda-Escudé & Rees (1997) discuss the possible detection of very high redshift core collapse SNe at [FORMULA] 5. By requiring that a metallicity [FORMULA] is produced at [FORMULA] 5, they estimate a rate of about one core collapse SN per square arcmin per year above [FORMULA] 5. Our extrapolated hierarchical model gives a rate of [FORMULA] 0.05 SN per square arcmin per year above [FORMULA] 5. This may favor models with a flat SFR at high z, which result in [FORMULA] 0.4 (0.7) SNe per square arcmin per year above [FORMULA] 5 when using [FORMULA] = 7 ([FORMULA] = 10). However, the metallicity used by Miralda-Escudé & Rees may be overestimated by an order of magnitude (Songaila 1997), leading to an overestimate of the SNR by the same amount. Also, the redshift before which the metallicity is assumed to have been produced affects the comparison. Using [FORMULA] 3, instead of [FORMULA] 5, decreases the estimated number SNe given by Miralda-Escudé & Rees by [FORMULA] 30%. More important, integrating our rates for redshifts above z = 3, instead of z = 5 as done above, results in a number of SNe that is a factor [FORMULA] 4 higher. Other uncertainties in the estimate by Miralda-Escudé & Rees include the actual fraction of the baryonic matter which is enriched by the SNe.

Miralda-Escudé & Rees limit their discussion to Type IIP SNe, and do not attempt a detailed discussion of the observed rates. The observed magnitudes compare fairly well with our magnitudes in the K and M bands, but are brighter in the optical and near-IR bands. The main reasons for this is that they use a higher effective temperature and that they do not take into account any line blanketing in the UV, as our models do. As we discuss in next section, the low metallicity may decrease this effect. Apart from these caveats, the discussion by Miralda-Escudé & Rees provides an important constraint at high redshifts.

Gilliland et al. (1999) report on the discovery of two high redshift SNe in the HDF (see also Mannucci & Ferrara 1999). One of the SNe has a probable host galaxy at [FORMULA] 0.95 (spectroscopically determined) and is possibly a Type II, whereas the other SN has a probable host galaxy at [FORMULA] 1.3 (photometrically determined) and is likely a Type Ia. Gilliland et al. also make detailed estimates of the expected number of Type Ia and Type II SNe in a HDF like search. With limiting magnitude IAB [FORMULA] 27.7 they find that [FORMULA] 0.32 Type Ia SNe should be detected in a search consisting of the HDF together with an observation of the same area made two years after the HDF. Using the same cosmological model as Gilliland et al. ([FORMULA] = 0.72, Ho = 63.3 km/s/Mpc3) and [FORMULA] = 1 Gyr, we estimate [FORMULA] 0.6 Type Ia SNe for a similar search. For a flat [FORMULA] = 1 cosmology we estimate 0.7 SNe. The main reason for the difference in results is that Gilliland et al. use a constant Type Ia SNR over the redshift range of interest (0 [FORMULA] 1.5), which is considerably lower than the mean value of our rates out to [FORMULA] 1.5.

For Type II SNe Gilliland et al. estimate [FORMULA] 1.2 SNe in HDF style search. This is in good agreement with our results, even though the modeling differs in many aspects. We estimate [FORMULA] 1.0 core collapse SNe for the cosmology used by Gilliland et al., and 1.3 SNe for a [FORMULA] = 1 cosmology.

Considering the small statistics, both estimates are consistent with the discovery of two SNe in the HDF.

Sadat et al. (1998) discuss the cosmic star formation rate, using a spectrophotometric model for different assumptions of the dust extinction. From this they calculate SNIa and core collapse rates, but do not translate these into directly observable rates. Their SFR is a factor [FORMULA] 3 higher than ours, which seems mostly to be due to the use of different factors when converting the observed luminosity densities to the SFRs. This also leads to higher SNRs (their Fig. 2). Sadat et al. also presents a case for Type Ia rates with a different normalization. These rates (their Fig. 3) agree better with our estimates at low redshifts. At high redshifts the Type Ia rates differ more due to different modeling of these SNe.

Jorgensen et al. (1997) attempt a calculation of the absolute rates of Type Ia, II and Ib SNe from a population model. Although in principle appealing, this model depends on the uncertain scenarios for the progenitors of especially the Type Ia's, as we have already discussed in this paper. Any estimates will therefore be sensitive to these assumptions. They also neglect the distinction between Type IIP's and Type IIL's, which most likely originate from different progenitors. Further, Jorgensen et al. assume in the calculation of the observed magnitudes in the different bands as function of redshift, that the spectrum is characterized by that at the peak. As we have discussed, the spectrum and luminosity vary strongly with time. The most serious deficiency is in our view their neglect of the magnitude variation, as given by the light curve, which as we have seen, changes the observed rates by large factors. Their estimates of the observed rates are therefore highly questionable.

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Online publication: October 4, 1999