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

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3. Results

Using the model above we can now estimate the observed number of SNe per square arcmin down to some limiting magnitude, including corrections from extinction and the shift of the spectrum with redshift.

As an illustration, we show in Fig. 2 the peak magnitudes (i.e. at [FORMULA] 10-15 days after explosion) for Types IIP, IIL, IIn and Ia as a function of redshift. We also show the magnitude of a Type IIP at the plateau phase (age [FORMULA] 40 days). The magnitude for Type Ib/c's at peak follows the Type Ia's, except for an off-set of [FORMULA]m [FORMULA] 1.5 mag towards fainter magnitudes. The magnitude of SN1987A-like SNe at the peak resembles the Type IIP at the plateau, but with an off-set [FORMULA]m [FORMULA] 2.0 towards fainter magnitudes. Besides the standard flat [FORMULA] = 1 cosmology (hereafter SCDM), we also show the apparent magnitudes for an open cosmology (OCDM) with [FORMULA] = 0.3 and [FORMULA] = 0, and a flat, [FORMULA]-dominated cosmology ([FORMULA]CDM) w ith [FORMULA] = 0.3 and [FORMULA] = 0.7. Note that the curves have a dispersion in magnitude according to values given in Table 1. The dispersion is largest for the Type IIP's; the peak magnitude of these may vary by more than one magnitude. Fig. 2 shows that the Type Ia's and the Type IIP's at the plateau drop out of the I and K´-band at [FORMULA] 1 and [FORMULA] 4.5, respectively, as a result of their UV-cutoffs. In contrast, the UV-bright Type IIL's and IIn's stay relatively bright in the I band even at high z.

[FIGURE] Fig. 2. Magnitudes as a function of redshift for different SN types in the I-band (left panels) and the K´-band (right panels). Top panels show SCDM ([FORMULA] = 1, [FORMULA] = 0), middle panels show OCDM ([FORMULA] = 0.3, [FORMULA] = 0) and lower panels show [FORMULA]CDM ([FORMULA] = 0.3, [FORMULA] = 0.7). Dotted line shows Type IIP at peak while the dash-dotted line shows the Type IIP at the plateau [FORMULA] 40 days after the explosion. Dashed, dash-triple-dot and solid lines show Type IIL, Type IIn and Type Ia, respectively. The magnitude for Type Ib/c's at peak follows the Type Ia's, except for an offset of [FORMULA]m [FORMULA] 1.5 mag towards fainter magnitudes. The magnitude of 1987A-like SNe resembles the Type IIP at the plateau, but with an offset [FORMULA]m [FORMULA] 2.0 towards fainter magnitudes. 

3.1. Core collapse rates

The solid lines in Fig. 3 show the number of predicted core collapse SNe per square arcmin in the R, I, K´ and M´ filters for different limiting AB-magnitudes. Because of the drop in the UV flux, bands bluer than R are of less interest for high redshifts.

[FIGURE] Fig. 3. Number of SNe per square arcmin that can be detected down to different limiting magnitudes in the R, I, K´ and M´ bands. Solid lines represent core collapse SNe. Dashed, dotted and dash-dotted lines represent Type Ia SNe with time delays [FORMULA]=0.3, 1 and 3 Gyr, respectively. Note that AB-magnitudes are used.

According to the specifications, NGST should have a detection limit of [FORMULA] 1 nJy in the J and K´ bands and [FORMULA] 3 nJy in the M´ band for a 104 s exposure with S/N=10 and [FORMULA] = 3 (Stockman 1997). These fluxes correspond to AB magnitudes of JAB = 31.4, [FORMULA] = 31.4 and [FORMULA] = 30.2, respectively. Using these limits, we find that the K´ band yields the highest number of core collapse SNe. In a 4[FORMULA]4 square arcmin field we predict [FORMULA] 45 simultaneously detectable core collapse SNe, with a mean redshift [FORMULA] = 1.9.

In Fig. 4 we show the redshift distribution in all bands for mAB = 26 and mAB = 31. For the lower magnitude limit one clearly sees the advantage of the infrared J, K´ and M´ bands when it comes to detect SNe with [FORMULA] 1. This is, of course, even more pronounced for mAB = 31, with the M´ band having the highest number of SNe at [FORMULA] 2. Note, however, that reaching RAB [FORMULA] 26 is considerably easier than JAB or [FORMULA] from ground. The smaller detector size in the latter bands is also a severe limiting factor.

[FIGURE] Fig. 4. Redshift distribution of core collapse SNe in the R, I, J, K´ and M´ filters for limiting magnitudes mAB=26 (left panel) and mAB=31 (right panel).

For SNe with redshifts [FORMULA] 5 we estimate [FORMULA] 1 SN per NGST field in the M´ filter. The actual numbers of the high-z SNe are highly uncertain, since the SNR at [FORMULA] is based on an extrapolation from lower redshifts. Also gravitational lensing may be important at these redshifts (see Sect. 6). However, independent of the actual numbers, we find that the M´ filter yields a factor [FORMULA] 2-5 higher counts compared to the K´ filter at these redshifts. We also find that NGST should be well suited to detect SNe originating from a possible Pop III at redshifts [FORMULA] 10.

The R and I bands sample light with rest wavelengths short-ward of the peak in the blackbody curves at lower redshifts than the J, K´ and M´ bands do. Besides the drop in luminosity due to the spectral shape (an effect especially pronounced for SNe with strong UV blanketing), these wavelengths are affected by larger extinction. The large K-correction decreases the rates in the R and I bands, and few SNe with [FORMULA] 1 are detected in I and R, even for limiting magnitudes [FORMULA] 29. Increases in these bands are instead caused by sampling SNe with fainter absolute magnitudes at [FORMULA] 1.

In Fig. 5 we show the redshift distribution of core collapse and Type Ia SNe down to IAB = 27 and [FORMULA] = 31, respectively. The total number of core collapse SNe is shown as the solid lines in each panel. The lower solid line shows the fraction of these coming from Type IIL and IIn SNe, which lack UV-cutoff in their spectrum over the whole light curve. The IAB = 27 panel clearly demonstrates the dominance of these at high redshift. Besides these, only the fraction of Type IIP's which are seen near the peak, before UV blanketing sets in, contribute to the SNe with [FORMULA] 1.5. The K´ band is far less sensitive to this effect, since the UV cutoff does not affect this filter at [FORMULA] 4.5. When we compare the redshift distribution for K´ = 27 and K´ = 31, we note that the mean redshift does not change much. However, the absolute number increases by a factor [FORMULA] 5. This is mainly caused by SNe below the light curve peak. Also the number of high redshift SNe with [FORMULA] 2 increases by a large factor.

[FIGURE] Fig. 5. Top panels: Observed SN distribution in redshift in the I filter, for limiting magnitudes IAB=25 and IAB=27. Upper solid lines show the total number of core collapse SNe, while the lower solid lines show the part originating from Type IIL and Type IIn SNe, i.e the SNe without a UV cutoff in their spectra over the whole light curve. Dashed, dotted and dash-dotted lines show the number of Type Ia SNe for models with [FORMULA]=0.3, 1 and 3 Gyr, respectively. Bottom panels: Same for K´ filter with limiting magnitudes [FORMULA] and [FORMULA]. Note the expanded redshift scale in the K´ band, as well as the different scales of the y-axis.

Table 2 gives some examples of our estimates for the number of SNe for NGST, VLT/FORS and HST/WFPC2, all for an exposure of 104 s and S/N = 10. For VLT/ISAAC and HST/NICMOS the small field and the relatively bright limiting magnitudes, compared to NGST, do not result in more than [FORMULA] 0.01 SNe per field. These instruments are therefore of limited interest when it comes to detecting high-z SNe.


[TABLE]

Table 2. Estimated number of SNe per field for different instruments. The limiting magnitudes are given for a 104s exposure and S/N=10.


So far we have discussed the number of SNe that are simultaneously observable during one search. To actually detect the SNe, additional observations are obviously required. Preferentially, a series of observations of each field should be undertaken in order to obtain a good sampling of the light curves. This also leads to the detection of new SNe in the additional frames. The number of new SNe depends on the total length of the search, and the spacing in time between each observation. As an illustration, in an idealized situation where a field is covered continuously during one year, the detected number of SNe per square degree, with limit IAB = 27, is increased from [FORMULA] 260 in the first image, to [FORMULA] 1650 for the whole years coverage. This procedure is, of course, observationally unrealistic. Using 80 days between each observation (i.e. [FORMULA] five observations for a years coverage) results in a total of 1150 different SNe. Using 40 days instead ([FORMULA] ten observations) gives [FORMULA] 1400 different SNe. Due to the time dilatation factor [FORMULA], one observes relatively fewer new SN explosions at high redshifts compared to the first detection.

With NGST, using limits as above, we estimate [FORMULA] 45 SNe in each frame. The total number of different SNe in a field that is covered continuously during one year is [FORMULA] 68. This means that in addition to the [FORMULA] 45 SNe observed in the first field, only [FORMULA] 23 new SNe have exploded during the year. With three observations and 180 days between the observations, [FORMULA] 63 different SNe are detected.

Compared to earlier estimates, our use of complete light curves during the whole evolution, as well as distribution of the SNe over time, results in a larger number of SNe, as well as a realistic distribution over redshift for a given magnitude. For example, with the same SFR and extinction, Madau et al. (1998a) predict [FORMULA] 7 SNe per NGST field per year in the range 2 [FORMULA] 4. Our calculations result in [FORMULA] 22. The difference is due to the fact that we use light curves covering the whole evolution, which allow us to include SNe at all epochs, instead of only those at peak, which is the case in Madau et al.

3.1.1. Shock breakout supernovae

Chugai et al. (1999) have noticed that the short peak in the light curve connected with the shock breakout may give rise to a transient event with a duration of a few hours. The possibility to observe this was pointed out already by Klein et al. (1979), although they concentrated mainly on the soft X-ray range.

Based on a radiation-hydrodynamics code, similar to that of Eastman et al. (1994), Chugai et al. have calculated monochromatic light curves for a Type IIP SN (or rather a scaled SN 1987A model) and a Type IIb (specifically SN1993J). With a short time interval between observations these SNe will be easily distinguishable from the Type Ia and Ib/c SNe, which have a rise time of t [FORMULA] 20 days (at [FORMULA] 1), and therefore only show a modest change in luminosity. A major problem in comparing their results to ours is that it is not discussed how they obtain their adopted intrinsic SNR's, although they approximately agree with those used in this paper, as well as Madau (1998). They also neglect dust extinction in their calculations.

Using this model, Chugai et al. find that two deep exposures, separated by [FORMULA] 10 days, result in 1.3 Type II SNe in the 6.8[FORMULA]6.8 square arcmin field of the VLT/FORS camera, using limit IAB = 28.2. In the calculations Chugai et al. assume that all SNe with [FORMULA] 2 are detected with this limit.

Using our hierarchical model, which gives approximately the same SNR up to [FORMULA] 2, and the same observational set-up, as Chugai et al., our calculations result in [FORMULA] 0.27 SNe. The reason for our lower estimate is that Chugai et al. assume all Type II SNe to have the same steep initial rise as the Type IIP and IIb. In our model we do not include any shock breakout for Types IIL and IIn, since the early time behavior of these SN types is not well known. If we do include all Type II SNe, our estimate increases to 0.66 SNe. The remaining discrepancy is mainly caused by the simplification Chugai et al. do by assuming that all SNe with [FORMULA] 2 are detected, combined with the fact that they do not include dust extinction in their calculations. Nevertheless, this may be an interesting way of studying the shock breakout of SNe. Unfortunately, it may be difficult to estimate the temperature and luminosity separately from this type of observations, because most of the observed evolution will be in the Rayleigh-Jeans part of the spectrum. Soft X-rays, as proposed by Klein et al., is here a better probe.

3.2. Type Ia SNe

Fig. 3 shows that in the R and I bands the number of core collapse SNe is comparable or larger than the number of Type Ia SNe for magnitudes fainter than [FORMULA]. The exact crossing point depends on the life time of the SN Ia progenitors, as well as the Type Ia normalization at low z. In the K´ and M´ bands core collapse SNe tend to dominate at all magnitudes. The reason for this difference is that the Type Ia SNe have a higher effective temperature over a longer period than the Type II SNe. The optical to IR flux ratio is therefore higher for the Type Ia's.

To illustrate the dependence on the progenitor life time, Fig. 3 gives the number of Type Ia SNe for [FORMULA] = 0.3, 1 and 3 Gyr in the different filters. Fig. 3 shows that a change in the progenitor life time, [FORMULA], introduces a non-negligible variation in the predicted number of observable SNe. For example, with an NGST detection limit, [FORMULA] = 31.4, we predict [FORMULA] 8 SNe for the two low values of [FORMULA], and [FORMULA] 5 for [FORMULA] = 3 Gyr. Observations in the I band with limits and field as for the VLT/FORS (see Table 2) results in 0.8, 1.1, and 1.8 SNe for increasing values of [FORMULA]. Counts may therefore seem like a useful probe to distinguish between different progenitor models (Ruiz-Lapuente & Canal 1998). However, the uncertainty in the modeling, especially in the normalization of the Type Ia rates to the local value, makes the counts highly model dependent. In next section we show that this is further hampered by the additional dispersion introduced when considering alternative star formation and extinction models.

If both SN type and redshift information are available for the observed SNe, it may be possible to use the redshift distribution of the SNe to distinguish between progenitor scenarios. Figs. 1 and 5 show that the peak in the SNR moves to lower z as [FORMULA] increases. Also, the high redshift cutoff in the rates depends strongly on [FORMULA]. Fig. 1 shows that the rates decrease towards zero at redshifts [FORMULA] 5.5, [FORMULA] 3.5 and [FORMULA] 1.5, for models with [FORMULA] = 0.3 Gyr, [FORMULA] = 1 Gyr and [FORMULA] = 3 Gyr, respectively. To reach these redshifts, filters unaffected by the UV cutoff must be used (i.e., [FORMULA] 4000(1+z) Å). Fig. 5 shows that the I filter is insensitive to Type Ia's at [FORMULA] 1.5, due to the spectral cutoff. Using the K´ filter (lower panels of Fig. 5) makes it possible to sample all types of SNe up to [FORMULA] 5, which is therefore most suitable for distinguishing different progenitor models. As already mentioned, this requires a determination of both SN type and redshift. A discussion of methods and problems regarding this follows in Sect. 7.

The only observational estimate of a Type Ia SNR at moderate redshift is by Pain et al. (1996). From a careful analysis, using realistic light curves and spectra, they find a Type Ia rate at [FORMULA] 0.4 of [FORMULA] SNe yr-1 deg-2 for 21.5 [FORMULA] RAB [FORMULA] 22.5. Using the same magnitude interval, and counting the SNe exploding during one year, we find [FORMULA] SNe yr-1 deg-2 for [FORMULA] = 0.3-3 Gyr, where the lower number corresponds to the [FORMULA] = 0.3 Gyr model. The mean redshift of these SNe is in our calculation [FORMULA] 0.35. These results seem to agree well, possibly favoring a high value of [FORMULA]. Note, however, that our estimated rate of Type Ia's is highly dependent on the normalization set by the local rate of these SNe (i.e., the efficiency parameter [FORMULA]). We have already seen that the uncertainty in this normalization may be a factor [FORMULA] 3.

3.2.1. Number of pre-maximum Type Ia SNe

The number of simultaneously detectable SNe discussed above is a result of events over the whole light curve. Using Type Ia's as standard candles for determination of [FORMULA] requires observations at the peak of the light curve, i.e that a first detection is made at the rising part of the light curve. We estimate the number of such SNe by assuming that the comoving rise time is 15 days. Fig. 6 shows the number of Type Ia SNe down to different limiting magnitudes before the peak of the light curve in the I filter for the three values of [FORMULA]. Also shown is the number of such SNe with [FORMULA] 1 (the lower sets of curves). An I band survey covering a one square degree field with limiting magnitude IAB [FORMULA] 27, will detect [FORMULA] pre-maximum Type Ia SNe (lower numbers for lower values of [FORMULA]), corresponding to [FORMULA] 30% of the total number. About [FORMULA] of the pre-maximum Type Ia's have redshifts [FORMULA] 1. These numbers are, again, sensitive to the local rate of Type Ia SN.

[FIGURE] Fig. 6. The number of simultaneously detectable Type Ia SNe on the rising part of the light curve for the I filter for three different values of [FORMULA]. Upper sets of curves show the total number of SNe, while the lower show SNe with [FORMULA]1.

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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