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Astron. Astrophys. 354, 767-786 (2000)

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7. Bias

The random magnification of distant sources by weak lensing induces some bias in any observed sample. For instance, the fraction of gravitationally lensed SNeIa increases for bright magnitudes due to the falloff of the luminosity function of the sources which means that the contribution from lower luminosity SNeIa which have been amplified by weak lensing becomes more important. We assume that the luminosity function of the SNeIa at redshift z is a gaussian, as a function of the absolute magnitude [FORMULA] :

[EQUATION]

Here [FORMULA] is the mean absolute magnitude while [FORMULA] is the intrinsic magnitude dispersion of the sources. Next, we define the bias [FORMULA] as the ratio of the number of SNeIa observed with a magnitude brighter than [FORMULA], at redshift z, by the number of SNeIa which actually are brighter than this luminosity threshold:

[EQUATION]

where [FORMULA] is the probability distribution of the magnification. Using (98) we obtain:

[EQUATION]

where we defined the "reduced magnitude" m by:

[EQUATION]

and [FORMULA] is the complementary error function. We can see from (99) that:

[EQUATION]

Indeed, for [FORMULA] one recovers all SNeIa (the survey is complete) while for [FORMULA] the local slope of the luminosity function is increasingly steep so that the effect of weak lensing gets larger.

We present in Fig. 11 our results for the source redshifts [FORMULA] and [FORMULA]. We can see that the bias is larger at higher redshift. Indeed, as can be seen from Fig. 3 the large µ tail of the probability distribution [FORMULA] is more important at higher z because of the larger variance [FORMULA]. This implies that the effect of weak lensing is higher (it is more likely for SNeIa to be "amplified" by density fluctuations) hence [FORMULA] is larger. Note that the bias is not negligible: we already get [FORMULA] for [FORMULA] at [FORMULA], which only corresponds to a "2 sigma" deviation from the mean [FORMULA] for the magnitude of the observed supernova.

[FIGURE] Fig. 11. The bias [FORMULA] for [FORMULA] (dashed lines) and [FORMULA] (solid lines).

These distortions due to weak lensing lead to an apparent luminosity function [FORMULA] of the sources:

[EQUATION]

Thus, the mean magnitude [FORMULA] of a survey limited by the absolute magnitude threshold [FORMULA] is:

[EQUATION]

[EQUATION]

The first term (with the exponential) mainly corresponds to the fact that the survey is limited by the upper magnitude [FORMULA] which implies that [FORMULA]. It vanishes for [FORMULA] when the survey is complete. The second term (with the prefactor [FORMULA]) does not go to zero even if [FORMULA]: it is due to the distortion of the luminosity function which implies that the mean of [FORMULA] is no longer [FORMULA]. Indeed, we obtain at the lowest order in [FORMULA]:

[EQUATION]

and:

[EQUATION]

As we can check in (105) the dispersion due to weak lensing increases the bright and faint tails of the luminosity function. However, this distortion is not symmetric, even at the lowest order in [FORMULA], because [FORMULA]. Hence the faint part of [FORMULA] shows a larger increase than the very bright part and [FORMULA]. As can be seen from (105) and Fig. 11 the effect of weak lensing is more important if the survey is not complete, hence the dependence on [FORMULA] (i.e. on the redshift [FORMULA] of the source) is larger. In particular, we have:

[EQUATION]

We present in Fig. 12 the deviation from the intrinsic mean [FORMULA] of the average magnitude measured by a survey with the upper absolute magnitude threshold [FORMULA], given by the "reduced magnitude" threshold [FORMULA] as defined in (100). For bright magnitude thresholds the apparent mean is close to [FORMULA] while for faint [FORMULA] the survey is almost complete and [FORMULA]. As explained above, even for [FORMULA] the observed mean is not equal to [FORMULA]. However, as we can see in (104) and in Fig. 12 the deviation is very small for the redshifts of interest [FORMULA] since the variance [FORMULA] of the magnification is small, see Fig. 1. In particular, we note that the curve [FORMULA] shows a very small dependence on redshift. In practice, in order to measure the cosmological parameters [FORMULA] and [FORMULA] one observes SNeIa at low ([FORMULA]) and high ([FORMULA]) redshift. The low [FORMULA] data gives the normalization of the curve [FORMULA], where [FORMULA] is the apparent magnitude, while the large [FORMULA] data constrains the cosmological parameters. Thus, the determination of the cosmological parameters is only sensitive to the difference between the deviations [FORMULA] at low and large [FORMULA]. Provided both surveys have a sufficiently faint absolute magnitude threshold: [FORMULA], the bias [FORMULA] due to weak lensing effects is:

[EQUATION]

where [FORMULA] are the redshifts of both surveys. Thus, if both surveys are almost complete we have:

[EQUATION]

For [FORMULA] we get [FORMULA], see Fig. 1. Thus this effect is negligible for the determination of cosmological parameters by SNeIa. Of course, if the low and large redshift surveys have bright magnitude thresholds [FORMULA] which are different it is not possible to estimate [FORMULA] nor [FORMULA]. This does not seem to be the case in practice (Perlmutter et al.1999).

[FIGURE] Fig. 12. The mean magnitude [FORMULA] of a survey limited by the apparent magnitude [FORMULA] for [FORMULA] (dashed lines) and [FORMULA] (solid lines). We note [FORMULA] the deviation of the threshold [FORMULA] from the intrinsic mean [FORMULA] in units of the dispersion [FORMULA].

In order to derive from observations the cosmological parameters [FORMULA] one draws a redshift [FORMULA] apparent magnitude diagram. Moreover, the apparent magnitude [FORMULA] is "corrected" thanks to the light-curve width-l uminosity relation (e.g. Perlmutter et al.1999). Of course, because there is some scatter in this latter relation there is still a small lightcurve-width-corrected dispersion [FORMULA] mag. The curve [FORMULA] depends on the cosmology, through the luminosity distance [FORMULA], which allows one to derive [FORMULA]:

[EQUATION]

where the observed apparent magnitude has been corrected for K and extinction corrections. However, due to the apparent magnitude threshold of SNeIa surveys a Malmquist bias appears at large redshifts where the mean apparent magnitude of supernovae is close to this threshold. Indeed, the curve one obtains from observations is not (109) but:

[EQUATION]

where [FORMULA] is the mean magnitude of a survey limited by the absolute magnitude threshold [FORMULA], as in (103). The latter is obtained from a given apparent magnitude threshold [FORMULA] by:

[EQUATION]

We display in Fig. 13 the curves [FORMULA] we obtain for three cosmologies (the term [FORMULA] removes the dependence on the Hubble constant). The dotted curves, which correspond to (109) (no magnitude threshold), show the (small) dependence on cosmology of the apparent magnitude-redshift relation: low-density universes lead to larger luminosity distances hence to larger apparent magnitude (fainter object) at fixed [FORMULA], this effect is larger for the flat model than for the open case. The solid (resp. dashed) curve shows the effect of an apparent magnitude threshold when weak gravitational lensing is (resp. is not) taken into account. The threshold is chosen so that [FORMULA] at redshift [FORMULA]. Thus, we see in the figure that at lower redshifts [FORMULA] the apparent magnitude threshold of the survey plays no role: all curves superpose onto (109). At higher redshift [FORMULA], because of the threshold [FORMULA], the survey only detects the brightest SNeIa which means that the average [FORMULA] is biased towards small magnitudes (large luminosities). This leads to a clear deviation of the observed magnitude-redshift relation from (109). Thus, the break in the curve clearly marks the redshift beyond which the cosmological parameters cannot be derived from observations with this apparent magnitude threshold. Taking into account the weak lensing effects (solid lines) slightly amplifies this bias towards large luminosities because the random magnification by density fluctuations along the line of sight increases the large apparent luminosity tail of the SNeIa distribution.

[FIGURE] Fig. 13. The redshift [FORMULA] apparent magnitude diagram for three cosmologies. We display the quantity [FORMULA] as a function of the redshift [FORMULA] of the source. The dotted curves show the apparent magnitude-redshift relation (109) obtained for a survey which is not limited by a flux threshold. The solid (resp. dashed) curves show the effect of an apparent magnitude threshold when weak gravitational lensing is (resp. is not) taken into account.

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Online publication: February 25, 2000
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