Astron. Astrophys. 362, 419-425 (2000)

5. The detection efficiency and the supernova rate

The supernova rate in the rest frame is expressed in SNu, i.e. in supernovæ per unit time per unit blue luminosity, . The number of supernovæ expected to be detected is given by a sum over the observed galaxies i weighted by their blue luminosities :

where is the efficiency to detect in galaxy i a type Ia supernova whose maximum occurs at time t in the supernova rest frame. The efficiency clearly depends on the galaxy redshift but also on other factors such as the galactic luminosity and surface-brightness profile. The integral is sometimes called the "control time" for the galaxy i.

We now need to determine the sum so the rate can be evaluated through Eq. 3. We do not know the luminosities of most of the observed galaxies because most of their redshifts are not known. As discussed in the previous section, we do, however, know the probability that the galaxy of magnitude has a redshift z. We therefore replace the above expression for with

where the blue luminosity is computed using the magnitude and the redshift z. The double integral in this equation was evaluated for each galaxy by Monte Carlo simulation of supernovæ that are subjected to the same detection procedures as the real data.

Because there is a better match between the EROS visible band and the standard V filter than with the standard B filter, the absolute luminosities of galaxies are first computed in the standard V band, and then converted into absolute luminosities in the standard B band. We use the difference between the solar colour: and a mean colour for galaxies, computed from the results of Fioc & Rocca-Volmerange (1997): .

5.1. Supernova simulation

The Monte Carlo simulation proceeds as follows. For each potential host galaxy, a series of redshifts is drawn randomly according to the distribution appropriate for the galaxy apparent magnitude. For each redshift, a time t since maximum is drawn uniformly between -15 and days in the supernova rest frame. The absolute magnitudes and of the supernova at that time are taken from the templates in these standard filters provided by Riess et al. (1996). To account for the observed spread of supernova luminosities, a gaussian scatter was added. The apparent magnitude of the simulated supernova is computed from the redshift using the K-correction from Nugent & Kim (in preparation) and assuming as above.

The supernova standard apparent magnitudes are then converted into a flux using our calibration. The position of the supernova in the host galaxy is drawn according to a two-dimensional Gaussian distribution, the first and second order moments and orientation of which are those of the host object.

As supernovæ are stellar sources, they appear on the frame with the same PSF as the field stars do. The PSF is modeled by a two-dimension integrated Gaussian. Its second order moments are set to the mean moments computed by fitting the stars on the same CCD quadrant, in order to take into account the spatial variation of the PSF.

5.2. Detection efficiency

The detection efficiency is derived by comparing the list of simulated supernovæ and the list of the supernovæ selected by the search cuts.

The detection efficiency as a function of magnitude is shown in Fig. 3. It appears as a smoothed step function, corresponding to a limiting magnitude . There is a slight dependence on the distance of the supernova from the host galaxy core, as shown in Fig. 4.

Fig. 3. The detection efficiency as a function of the simulated supernova magnitude at discovery. It corresponds to a limiting magnitude . The maximum efficiency is below 1 because of the masked areas near CCD defects and saturated stars.

Fig. 4. Dependence of the detection efficiency on the distance between the simulated supernova and its host galaxy core. The average efficiency is as low as 30% because supernova light curves are sampled up to 120 days after maximum, causing % of them to drop below .

As foreseen, the detection efficiency depends on the host galaxy characteristics, such as the surface brightness or the local gradient where the simulated supernova is added. These results are presented in Fig. 5, where the distribution of these two parameters have been superimposed on the efficiency. The distributions are shown for supernovæ whose magnitude is in the range . They peak where the efficiency is good, thus moderating the impact of the efficiency behaviour.

Fig. 5. Detection efficiency as a function of the host galaxy surface brightness (left pannel) and of the local gradient at the point where the simulated supernova is added (right pannel). The local gradient was taken to be where B is the local Poisson noise. The distributions of the surface brightness and the local gradient are superimposed (dashed lines). The distributions are shown for supernovæ whose magnitude is in the range .

5.3. Redshift distribution

The Monte-Carlo computed redshift distribution of detected supernovæ is shown in Fig. 6. The thin solid line shows the distribution when all the analysis cuts are applied. For comparison, the dotted line shows the redshift distribution when only two cuts are applied, one on the supernova magnitude , and one on the host galaxy magnitude equivalent to . The thick solid line shows the distribution obtained with an analytic model where we assume that the type Ia supernova rate is proportionnal to the host galaxy luminosity and that the galaxy luminosity distribution is given by the LCRS Schechter law (Eq. 2). In this analytic model the cuts are described by two step-functions whose thresholds are respectively and . Using this analytic model, we find that imposing the host galaxy magnitude cut reduces the number of detected supernova by a factor .

Fig. 6. The expected redshift distribution. The thin solid line shows the distribution in z derived from the Monte-Carlo simulation when the analysis cuts are applied. The dotted line shows the same but with only two cuts applied, one on the supernova magnitude () and the other one on the host galaxy magnitude equivalent to . The latter distribution compares well with a simple analytic model as described in the text (thick solid line). The arrows indicate the redshift values of the 4 discovered type Ia supernovæ.

The observed redshift distribution of the 4 discovered type Ia supernovæ is in agreement with the Monte-Carlo distribution (the Kolmogorov-Smirnov compatibility test gives a 25% probability), although the small-number statistics prevents any definitive conclusion. The mean and rms of the expected redshifts, respectively 0.14 and 0.06, are to be compared with the mean and rms of the observed redshifts, respectively 0.10 and 0.06.

5.4. Explosion rate

The rate is computed by dividing the number of observed supernovæ by the sum :

The sum obtained is . For type Ia supernovæ, this corresponds to a rate where the error is statistical and given at a 68% confidence level.

© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000
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