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

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2. The model

2.1. Filters and magnitude system

Because of the strong UV deficiency in the spectrum of most SN types, filters bluer than R are of little interest for high-z studies. For R ([FORMULA] = 0.65 µm) and I ([FORMULA] = 0.8 µm) we use Cousins filters, and in the near-IR J ([FORMULA] = 1.2 µm) and K´ ([FORMULA] = 2.1 µm) filters. For the M band we use a filter that we denote by M´, which is centered on [FORMULA] = 4.2 µm with [FORMULA] = 3. For the magnitudes we use the AB-system where mAB = [FORMULA] 48.6 (Oke & Gunn 1983) ([FORMULA] in ergs cm-2 Hz-1 s-1). Here 1 nJy corresponds to mAB = 31.4. When needed, we have used the following relations to transform from Vega based magnitudes to AB magnitudes: K´ = [FORMULA], J = [FORMULA], [FORMULA] and R = [FORMULA].

2.2. Star formation rates

A problem when using UV-luminosity densities to calculate the SFR is that these can be matched to almost any SFR, ranging from a strongly peaked to a flat, or even increasing, SFR at [FORMULA] 1, by adjusting the assumed extinction due to dust. By simultaneously using luminosity densities observed in different wavebands it is possible to break this degeneracy. Madau (1998) used three bands (UV, optical and NIR) in order to derive a SFR that can simultaneously reproduce the evolution of the different luminosity densities. He found a best fit for a universal extinction [FORMULA]=0.1 with SMC-type dust. Despite corrections for absorption, the SFR still shows a pronounced peak at 1 [FORMULA] 2. This SFR is shown in Fig. 1. A peaked SFR is predicted in scenarios where galaxies form hierarchically (e.g., Cole et al. 1994). Although commonly used, this SFR is not universally accepted and in Sect. 4 we discuss alternative SFR's.

[FIGURE] Fig. 1. SNRs for the model with dust extinction [FORMULA]=0.1. The solid line represent the rate of core collapse SNe. This scales directly with the SFR, given by the right hand axis. Dashed, dotted and dash-dotted lines represent the rate of Type Ia SNe with times delays [FORMULA]=0.3, 1.0 and 3.0 Gyr, respectively. The rates of Type Ia SNe are normalized to fit the locally observed rates.

Core collapse SNe, which originate from short lived massive stars (ages [FORMULA] yrs), have an evolution that closely follows the shape of the SFR. Assuming an immediate conversion of these stars to SNe renders a multiplicative factor, k = SNR/SFR, between the SFR ([FORMULA] yr-1Mpc-3) and the SNR (yr-1Mpc-3), where k depends on the IMF and the mass range of the progenitors. Here, we take

[EQUATION]

Using a Salpeter IMF yields k=0.0064. Note that, for consistency, the same IMF as assumed when deriving the SFR from the luminosity densities should to be used. Using a Scalo IMF, instead of a Salpeter IMF, decreases the conversion factor k between the SFR and the SNR by a factor 2.6. A Scalo IMF, however, also increases the SFR as derived from the UV luminosities by a factor [FORMULA] 2, hence cancelling most of the effect. The reason is that the same stars that produce the UV luminosities also explode as core collapse SNe.

The lower mass limit for Type II progenitors is generally believed to be 8-11 [FORMULA] (e.g., Timmes et al. 1996). Here we choose 8 [FORMULA], a limit supported by e.g., Nomoto (1984). An increase to 11 [FORMULA] would decrease k, and hence the SNR, by [FORMULA] 38%. Indications from especially the oxygen/iron ratio that the most heavy stars form black holes and do not result in SNe, justifies the use of 50 [FORMULA] as an upper limit to the progenitor mass (Tsujimoto et al. 1997). Other studies (e.g., Timmes et al. 1995), however, find that stars with masses down to [FORMULA] 30 [FORMULA] may result in black holes. An upper limit of 30 [FORMULA], instead of 50 [FORMULA], decreases k by [FORMULA] 9%.

The uncertainties concerning the origin of Type Ia SNe makes the relation between the rate of these SNe and the star formation more ambiguous. The predicted SNR depends on the nature of the SN progenitors. Yoshii et al. (1996) argue that the SNe Ia progenitor lifetime is probably restricted to 0.5-3 Gyr. This range opens a possible way to observationally distinguish between progenitor models. Ruiz-Lapuente & Canal (1998) find that the more short-lived double-degenerate progenitor systems and the long-lived cataclysmic-like systems should yield significantly different rates.

In this paper we use as a standard model the SFRs derived by Madau (1998) to calculate the rates of both core collapse and Type Ia SNe. We use a universal extinction [FORMULA]=0.1, together with SMC-type dust and a Salpeter IMF. In Sect. 4 we investigate how a higher extinction affects the counts, and if it is possible to use these counts to estimate the amount of dust.

2.2.1. Core collapse SNe

The SNR used is shown in Fig. 1. This is the intrinsic rate of exploding SNe, irrespective of magnitudes and spectral distributions. The apparent magnitude of a SN at redshift z at a time t off its peak magnitude (i.e. t can be negative) in a host galaxy with inclination i observed in a filter f is given by

[EQUATION]

Here, [FORMULA] is the absolute magnitude of the SN in filter f at time t relative to the peak of the light curve, [FORMULA] is the distance modulus,

[EQUATION]

The luminosity distance, [FORMULA], is given by,

[EQUATION]

where [FORMULA] and `sinn' stands for [FORMULA] if [FORMULA] 0 and for [FORMULA] if [FORMULA] 0 (Misner et al. 1973). If [FORMULA] = 0 then the `sinn' and the [FORMULA] terms are set equal to one. For the standard CDM cosmology mainly used here, the distance modulus is given by [FORMULA]= 45.4 - 5log[FORMULA] km s-1 Mpc-1) + 5log[(1+z) - [FORMULA]]. Further, [FORMULA] gives the K-correction, [FORMULA] is the Galactic absorption, and [FORMULA] is the radially averaged absorption in the parent galaxy with inclination i.

For Type Ib/c, plateau Type IIP and linear Type IIL SNe we use peak magnitudes given by Miller & Branch (1990). The magnitudes of the faint SN1987A-like SNe are not well known. Here we adopt the magnitudes given by Cappellaro et al. (1993), while magnitudes from Patat et al. (1994) are adopted for the Type IIn SNe. The magnitudes given by these authors are, however, not corrected for absorption. Adopting an average [FORMULA] = 0.1 yields a mean [FORMULA] = 0.41 mag. Taking the effect of the albedo of the dust grains into account lowers the effective absorption. We adopt an absorption [FORMULA] = 0.32 mag, which is consistent with the mean face-on absorption calculated by Hatano et al. (1998) in their models for Type II SN extinction. We have here assumed that the absorption of the light from the parent galaxy and the SNe follow the same extinction law, implying that the SNe and the progenitor stars occur in the same environment, which to some extent may be incorrect. If the first core collapse SNe and their main sequence progenitors in a region sweeps away part of the shrouding dust, the absorption could be lower than calculated above. In Sect. 4 we return to this possibility. In Table 1 we list the corrected mean absolute B magnitudes, as well as the adopted dispersion in this quantity for the different types at the peak of the light curve.


[TABLE]

Table 1. Adopted maximum absolute B magnitudes and dispersion in the AB-system. From Miller & Branch (1990), Cappellaro et al. (1997), Cappellaro et al. (1993) and Patat et al. (1994). Values are corrected for extinction [FORMULA]=0.1, corresponding to [FORMULA]=0.32. The intrinsic fraction of exploding SNe of the different types are given by f.


For the evolution of the luminosity with time we use light curves from Filippenko (1997) for the Type Ib/c's, IIP's, IIL's and the SN1987-like SNe. We have made some modifications to the light curve for the IIP's to achieve a better fit to the observed, as well as theoretical, light curves presented in Eastman et al. (1994). For Type IIn's we use a light curve intermediate between IIP and IIL. This seems adequate in view of the data presented by Patat et al. (1994), who find that Type IIn SNe can have both linear and plateau shape, but there are also light curves in-between these two.

The K-correction is calculated by assuming modified blackbodies for the spectral distributions of the SNe. At [FORMULA] 1 even the I-band corresponds to the rest UV part of the spectrum. The exact spectral distribution in the blue and UV is therefore crucial. Unfortunately, this part of the spectrum is relatively unexplored even at low z. UV observations of Type IIP's are especially scarce. Only for the somewhat peculiar Type IIP SN1987A is there a good UV coverage (Pun et al. 1995). This showed already [FORMULA] 3 days after explosion a very strong UV deficit, similar to Type Ia SNe. This is a result of the strong line blanketing by lines from Fe II, Fe III, Ti II and other iron peak elements (e.g., Lucy 1987; Eastman & Kirshner 1989). Although the progenitors of typical Type IIP's probably are red supergiants, rather than blue as for SN1987A, there is no reason why these ions should be less abundant than in SN1987A. On the contrary, because of the near absence of strong circumstellar interaction and similar temperature evolution they are expected to have fairly similar UV spectra, as calculations by Eastman et al. (1994) also show. Eastman et al. find that the UV blanketing sets in after [FORMULA] 20 days when the effective temperature becomes less than [FORMULA] 7000 K. This coincides with the beginning of the plateau phase. We therefore mimic the UV blanketing by a 2 magnitude drop between 4000 Å and 3000 Å, and a total cutoff short-wards of 3000 Å, for the Type IIP and SN1987A-like SNe with Teff [FORMULA] 7000 K. At higher Teff we assume non-truncated blackbodies for these types. The time evolution of the spectra is modeled by changing the characteristic temperature of the blackbody curves to agree with the calculations by Eastman et al. At shock outbreak a short interval of energetic UV radiation occurs with [FORMULA] [FORMULA] Teff [FORMULA] 106 K. This only lasts a few hours, but may due to its high luminosity be observable to high z (Klein et al. 1979; Blinnikov et al. 1998; Chugai et al. 1999). It then cools to [FORMULA] 2.5[FORMULA] K at [FORMULA] 1 day and further to [FORMULA] 7000 K after [FORMULA] 20 days. At the plateau phase the temperature is [FORMULA] 5000 K.

From the limited UV information of the Type IIL's SN 1979C, SN 1980K and SN 1985L (Cappellaro et al. 1995), and the Type IIn SN 1998S (Kirshner et al., private communication), we model the spectra of the IIL's and IIn's by blackbody spectra without cutoffs. The most extensive coverage of the UV spectrum of a Type IIL exists for SN 1979C (Panagia 1982). By fitting blackbody spectra to the optical photometry, we find that even after two months, when the color temperature is only [FORMULA] 6000 K, there is no indication of UV blanketing. On the contrary, there is throughout the evolution a fairly strong UV excess, consisting of continuum emission as well as lines, in particular Mg II 2800 Å. The UV excess of these SNe is likely to be caused by the interaction of the SN and its circumstellar medium. This ionizes and heats the outer layers of the SN, decreasing the UV blanketing strongly. For the effective temperature as a function of time we use the fits for SN 1979C by Branch et al. (1981). The Type Ib/c light curve is taken from Filippenko (1997) and is assumed to have the same spectral characteristics as the Type Ia (described further below).

The Galactic absorption, [FORMULA], is taken to be zero in our modeling. This can easily be changed to other values of [FORMULA]. The term [FORMULA] adds the absorption due to internal dust in the parent galaxy with inclination i, according to the adopted extinction laws. Gordon et al. (1997) show that the extinction curve for starburst galaxies lacks the 2175 Å bump, like an SMC-type extinction curve does, and shows a steep far-UV rise, intermediate between a Milky Way and an SMC-like extinction curve. The observed increase with redshift of the UV-luminosity originates mainly from starburst/irregular systems (e.g., Brinchmann et al. 1998). This implies that a major part of the core collapse SNe should be found in such galaxies, and an SMC-type extinction curve should therefore be most appropriate when calculating the absorption of the SNe light. The difference between an SMC-type dust curve and a Milky Way-type extinction in the interesting wavelength range is small. This is especially true for SNe with a short wavelength cutoff in their spectral energy distributions, but also a blackbody spectrum with Teff [FORMULA] 7000 K drops fast enough at short wavelengths for the precise form of UV absorption in this region to be less important.

The dependence on inclination has been modeled by Hatano et al. (1998). The absorption closely follows a (cos i)-1 behavior up to high inclinations. We approximate their results by [FORMULA] = 0.32[(cosi)-1 - 1]. Also the radial dependence of the absorption is discussed by Hatano et al. We simplify our calculations by adopting their radially averaged value of the absorption. Dividing the host galaxies into different types increases the dispersion in absorption. Finally, the extinction may depend on the intrinsic luminosity and the metallicity of the host galaxy. This variation should to some extent already be accounted for in the observed dispersion of the peak magnitudes.

The SNe are divided into the five different groups, with fractions, f, representing the intrinsic fraction of exploding core collapse SNe of the different types (i.e. irrespective of magnitude). We estimate f by using the observed ratios of discovery , [FORMULA], given by Cappellaro et al. (1993), who find [FORMULA](IIP)[FORMULA](IIL), [FORMULA](Ib/c)[FORMULA] 0.3[FORMULA](II) and [FORMULA](IIn)[FORMULA] 0.2[FORMULA](II).

The total number of Type II's is the sum of Type IIP, IIL, 1987A-like and IIn. Adding the Type Ib/c yields the total number of core collapse SNe. Cappellaro et al. (1997) argue that the intrinsic fractions of IIn and 1987A-like should be f(IIn) = (0.02-0.05)f(II) and f(1987A-like) = (0.10-0.30)f(II). The IIP, IIL and Ib Types have approximately the same magnitudes, i.e. the ratio of discovery should be close to the intrinsic fractions between these. Combining these values and assumptions leads to our adopted intrinsic fractions in Table 1. We note here that, unfortunately, the rates of the different classes are affected by fairly large uncertainties.

The observable number of SNe with different apparent magnitudes is calculated by integrating the SNR over redshift. The SNe are distributed between the different types according to their intrinsic fractions, and are placed in parent galaxies with inclinations between 0o to 90o. An important feature of our model is that the number of SNe exploding each year are distributed in time, and are given absolute magnitudes consistent with their light curves. With this procedure we obtain the simultaneously observable number of SNe, including both those close to peak and those at late epoch. In order to actually detect the SNe, at least one more observation has to be made after an appropriate time has passed. In Sect. 8 we discuss the spacing in time between observations.

For [FORMULA] 5 we use a rate that is an extrapolation from lower z. This certainly simplifies the actual situation drastically. However, due to the large distance modulus and the decline in the rate at high z, the fraction of SNe with [FORMULA] 5 is small, and hence the errors due to the uncertainty in the shape of the SFR at [FORMULA] 5. Furthermore, SNe with a spectral cutoff at short wavelengths drop out at redshifts [FORMULA] -1, where [FORMULA] is the effective wavelength of the filter. In the R, I, J, K´ and M´ filters this occurs at [FORMULA] 0.65, 1.0, 2.0, 4.5 and 10, respectively. This makes the contribution from SNe with higher redshifts insignificant. A caveat here is that lensing, as well as an early epoch of Pop III SNe, may cause a significant deviation from this extrapolation. These issues are discussed in Sect. 4 and Sect. 6.

2.2.2. Type Ia SNe

When calculating the number of Type Ia SNe we employ the same procedure as above. We use an extinction corrected peak magnitude [FORMULA] = -19.99 and dispersion [FORMULA] = 0.27, found by Miller & Branch (1990). The time delay between the formation of the progenitor star and explosion of the SNe is treated in a way similar to Madau et al. (1998a). Most likely, the progenitors are stars with mass [FORMULA] (Nomoto et al. 1994). Stars forming at time [FORMULA] reach the white dwarf phase at [FORMULA], where [FORMULA]=10([FORMULA] Gyr is the time spent on the main sequence. After spending a time [FORMULA] in the white dwarf phase, a fraction [FORMULA] of the progenitors explode as a result of binary accretion at [FORMULA]. The SNR at time t can then be written

[EQUATION]

where [FORMULA] is the time corresponding to the redshift of the formation of the first stars, [FORMULA]. Arguments from Yoshii et al. (1996) and Ruiz-Lapuente & Canal (1998) indicate 0.3 [FORMULA] 3 Gyr. In the calculations we therefore use three different values for the time delay, [FORMULA] = 0.3, 1 and 3 Gyr. The [FORMULA] = 0.3 Gyr model approximately mimics the double degenerate case, while the [FORMULA] = 1 Gyr model resembles the cataclysmic progenitor model. With the additional [FORMULA] = 3 Gyr the range is expanded to cover all likely models.

The parameter [FORMULA] in Eq. (6) is introduced to give the fraction of stars in the interval [FORMULA] [FORMULA] that result in Type Ia SNe, and is determined by fitting the estimated SNR at z = 0 to the locally observed Type Ia rates. For an alternative approach based on specific assumptions about the progenitors see Jorgensen et al. (1997). Results from local SN searches (Cappellaro et al. 1997; Tammann et al. 1994; Evans et al. 1989) give local rates of 0.12, 0.19, and 0.12 SNu, respectively (1 SNu= 1 SN per century per [FORMULA]). Adopting a mean of 0.14 [FORMULA] 0.06 SNu (Madau et al. 1998a), and using a local B band luminosity estimated by Ellis et al. (1996), leads to a local Type Ia rate of 1.3 [FORMULA] 0.6 [FORMULA]10-5 SNe yr-1 Mpc-3. The normalization at z = 0 yields an efficiency 0.04 [FORMULA] 0.08, where the range is due to the fact that different values of [FORMULA] gives different [FORMULA] in order to reproduce the local rates. The uncertainty in the local SNR is equivalent to an additional spread in [FORMULA] by a factor [FORMULA] 3. The normalization to the local rate leads to an uncertainty in the Type Ia SNR at all redshifts that corresponds to the uncertainty in the local rate. This implies an uncertainty by a factor [FORMULA] 3 in the estimated Type Ia rates.

The treatment of absorption in the case of Type Ia SNe is more complicated than in the case of core collapse SNe, and is therefore subject to larger uncertainties. The long time delay between the formation of the progenitors and the explosion unties the environmental link between these events. A (binary) star forming in a dusty starburst region may e.g. explode much later as a Type Ia SN in a dust-free elliptical galaxy. Observations show that Type Ia SNe do occur in both elliptical and spiral galaxies. However, the fact that the major part of local Type Ia SNe are detected in spirals, together with observations that indicate that the global fraction of ellipticals, or at least low-dust ellipticals, seems to decrease at increasing redshifts (Driver et al. 1998) may justify our simplification of putting all the SNe Ia in spiral environments. Ignoring the elliptical parent galaxies leads to a slight overestimate of the absorption, and a slight underestimate of the SN detection rate.

Absorption in both the bulge and disk components of spiral galaxies have been calculated by Hatano et al. (1998). They find a somewhat smaller absorption for Type Ia SNe than for core collapse SNe. The Type Ia absorption is also less dependent on the inclination of the parent galaxy. We use the disk component results of Hatano et al. in our model for the absorption of the Type Ia SNe. Fig. 1 show the intrinsic Type Ia SNRs for the three time delays used. Increasing the time delay shifts the peak of the Ia's towards lower redshift. The SFR (and the rate of core collapse SNe) peak at [FORMULA] 1.55, while the Type Ia rates in the [FORMULA] = 0.3, 1 and 3 Gyr models peak at [FORMULA] 1.35, [FORMULA] 1.16, and [FORMULA] 0.71, respectively.

To describe the spectral energy distribution of the Type Ia SNe we use blackbody curves with a spectral cutoff at [FORMULA] 4000 Å (e.g., Branch et al. 1983). The temperature is set to 15 000 K around the peak, decreasing to 6000 K after [FORMULA] 25 days (e.g., Schurmann 1983). We have compared the K-corrections in our model to those calculated from real spectra by Kim et al. (1996). The mean deviation in the R-band correction between our modified black body curves and the detailed calculations by Kim et al. is [FORMULA] 0.1 mag. This agreement justifies the use of the blackbody representation for the SN spectra. The average Type Ia light curve is taken from Riess et al. (1999).

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Online publication: October 4, 1999
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