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

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6. Derivation of cosmological parameters

In practice, one uses the Type Ia supernovae as standard candles in order to derive the cosmological parameters [FORMULA] and [FORMULA] from the observed relation redshift [FORMULA] luminosity distance (e.g. Perlmutter et al.1999). However, even if these sources are perfect candles with no intrinsic dispersion nor instrumental noise, the weak lensing effects discussed in the previous sections will introduce some dispersion and some bias as the supernovae will be randomly magnified by the density fluctuations located along the line of sight. Thus, the luminosity distance measured by the observer will differ from the actual one by a small deviation which we can relate to the magnification µ by:

[EQUATION]

where [FORMULA] is the observed distance defined in (18) seen as a function of [FORMULA] and [FORMULA] which are thus determined by the observation (if there is no distortion by weak lensing: [FORMULA], these parameters are equal to the actual cosmological parameters [FORMULA] and [FORMULA]). Here we assume the absolute magnitude of the supernovae is known, for instance from local or low redshifts observations. We used the fact that the luminosity distance [FORMULA] obeys: [FORMULA]. Thus, for a given redshift [FORMULA] of the source and a peculiar cosmology [FORMULA] of the actual universe we can obtain "confidence regions" in the [FORMULA] plane for the observed parameters [FORMULA] and [FORMULA]. This means that for any [FORMULA] with [FORMULA], the "observed universe" has a probability [FORMULA] to lie within the domain [FORMULA]. Thus, for any point [FORMULA] we calculate through (94) the weak lensing magnification µ which is needed so that the observer would measure [FORMULA] in an actual universe defined by [FORMULA]. Then, from the intervals of confidence (60), displayed in Fig. 4, we obtain the probability [FORMULA] (defined by the constraint that either [FORMULA] or [FORMULA]) that such a magnification is realized, hence that such a cosmology is derived from observations. In this way we obtain confidence regions [FORMULA] in the [FORMULA] plane. Of course, from (94) a measure at a single redshift only provides a value for the distance [FORMULA] (hence for the deceleration parameter [FORMULA] at low redshift [FORMULA]). Hence the parameters [FORMULA] and [FORMULA] are only constrained to lie on a line in the [FORMULA] plane. As a consequence, the regions [FORMULA] are unbounded strips in the [FORMULA] plane.

We display in Fig. 9 these "confidence regions" [FORMULA] we obtain for the three cosmologies we have studied in details in the previous sections, for the redshifts [FORMULA] and [FORMULA]. At a given redshift [FORMULA], we show the domains [FORMULA] defined by [FORMULA] (dashed region within two solid lines) and by [FORMULA] (within the two dashed lines). This means that from one observation at a given redshift the observer will conclude with a probability of [FORMULA] that the cosmological parameters [FORMULA] lie in the dashed domain. We also display the boundary line [FORMULA] given by [FORMULA] within (94) (solid line on the left side). Thus, because of the lower bound [FORMULA] the observed parameters [FORMULA] cannot lie to the left of the line [FORMULA] (of course, here we only consider the effects of weak lensing). The cross is the point [FORMULA]. For a given redshift, all these boundary lines are parallel. Moreover, their slope (when seen as a function [FORMULA] of [FORMULA]) increases with [FORMULA]. Indeed, as emphasized by Perlmutter et al.(1997) the parameters [FORMULA] and [FORMULA] enter [FORMULA] with different powers of [FORMULA] so that the observed distance [FORMULA] is not a function of [FORMULA] (except in the limit [FORMULA]) but of a combination of [FORMULA] and [FORMULA] which varies with [FORMULA]. This implies the drift with [FORMULA] of the slope of the boundary lines we defined above. Of course, this is the reason why observations can simultaneously constrain [FORMULA] and [FORMULA] (provided one has a finite range of source redshifts). The sign of the slope of these strips translates the fact that for the same [FORMULA] and redshift a flat universe corresponds to a larger distance [FORMULA] than an open geometry. Of course, the strong asymmetry of the probability distribution [FORMULA], and of the intervals [FORMULA], leads to asymmetric regions [FORMULA]. Thus, as [FORMULA] increases the strip [FORMULA] grows but it is bounded on the left by [FORMULA] while it can extend to infinity to the right. Indeed, as can be seen from (94) larger µ corresponds to larger [FORMULA] and smaller [FORMULA]. Thus, the most likely values of [FORMULA], corresponding to [FORMULA], lie slightly to the left of the point [FORMULA], while for large [FORMULA] the domain [FORMULA] shows an extended tail towards large [FORMULA]. The strips are larger for the critical universe which had a higher dispersion for the probability distribution [FORMULA], see Fig. 3. We note that although the spread due to weak lensing cannot make a critical universe appear as [FORMULA] (due to the cutoff [FORMULA]) while a low-density universe [FORMULA] has a negligible probability to appear as [FORMULA], the effect of the weak lensing is not negligible. Thus, two observations at redshifts [FORMULA] and [FORMULA] only determine [FORMULA] within an interval [FORMULA]. For instance, for the low-density flat universe (lower panel) we have:

[EQUATION]

Note that if both observations (at [FORMULA] and [FORMULA]) are independent the intersection of the regions labelled [FORMULA], used in (95), corresponds to a probability [FORMULA]. These uncertainties are larger for both other cosmologies we consider in Fig. 9. Of course, by averaging over many observations at a given redshift one diminishes this inaccuracy (one expects that [FORMULA] roughly decreases as [FORMULA]). Moreover, we can check in the figure that observations can unambiguously discriminate between [FORMULA] and [FORMULA]. In the case of a low-density universe one can also clearly discriminate between [FORMULA] and [FORMULA].

[FIGURE] Fig. 9. The "confidence regions" [FORMULA] for the observed parameters [FORMULA] and [FORMULA], for three actual cosmologies and the redshifts [FORMULA] and [FORMULA]. The dashed region within two solid lines corresponds to [FORMULA]: within [FORMULA] of the cases the cosmology will be observed to lie within this domain in the [FORMULA] plane. The region within the two dashed lines corresponds to [FORMULA]. The solid line [FORMULA] on the left corresponds to the lower bound [FORMULA]: the observed parameters [FORMULA] cannot lie to the left of this curve. The boundary lines obtained for a larger redshift are steeper (i.e. closer to the vertical).

In order to compare the effect of weak gravitational lensing on the derivation of the cosmological parameters with the inaccuracy due to the intrinsic magnitude dispersion [FORMULA] of the sources, we show in Fig. 10 the quantity:

[EQUATION]

[FIGURE] Fig. 10. The uncertainty [FORMULA] of the observed parameter [FORMULA] from sources at redshift [FORMULA], for three cosmologies. The solid lines take into account both the intrinsic dispersion [FORMULA] of supernovae and the fluctuation [FORMULA] of the magnification by weak lensing. The dashed curves show the influence of [FORMULA] alone (case with [FORMULA]) while the dotted curves represent the influence of weak lensing alone (case with [FORMULA]). For an actual universe which is critical ([FORMULA]) the lower curves correspond to the assumption of a flat universe and the upper curves to [FORMULA] universes. For low-density universes we assume the correct cosmology (flat or open).

Here [FORMULA] is the absolute magnitude of the supernova and [FORMULA] is the magnitude dispersion due to weak lensing. Thus, we have:

[EQUATION]

where [FORMULA] is obtained from (94). For low-density universe we eliminate [FORMULA] in (94) by assuming the right cosmology (flat or [FORMULA]) so that [FORMULA] is a function of [FORMULA] al one. In the case of a critical universe we display the results we obtain when we assume [FORMULA] or [FORMULA]. We use for all redshifts [FORMULA] mag for the lightcurve-width-corrected luminosity dispersion of SNeIa, see Perlmutter et al.(1999). The solid lines in Fig. 10 show [FORMULA] from (97). The dashed curves show the inaccuracy on [FORMULA] due to [FORMULA] alone while the dotted curves show the effect o [FORMULA] alone. We can see that for low-density universes the error due to the intrinsic dispersion [FORMULA] dominates up to [FORMULA] while for a critical universe the weak lensing contribution already dominates for [FORMULA]. Note also that the inaccuracy of the measure of [FORMULA] is much larger for a critical universe, which has a non-negligible probability to appear as an open universe with [FORMULA] or a flat universe with [FORMULA] for [FORMULA]. In particular, note that going to high redshift [FORMULA] does not increase the accuracy of the determination of [FORMULA] by much. Thus, the minimum dispersion of [FORMULA] is [FORMULA] for low-density universe and [FORMULA] for a critical universe. Of course, one can reduce the uncertainties by observing many supernovae. Note that this analysis does not take into account the non-gaussian behaviour of the probability distribution of the magnification µ. This was studied in Fig. 9. Moreover, as can be seen in Fig. 9 the uncertainty on [FORMULA] due to weak gravitational lensing is larger than the estimate (97) because of the degeneracy in the plane [FORMULA]. Indeed, although observations at two different redshifts [FORMULA] and [FORMULA] remove this degeneracy the intersection of the domains [FORMULA] remains elongated along an axis roughly parallel to [FORMULA]. Hence the uncertainty on [FORMULA], given by the projection onto the [FORMULA]-coordinate of the length of this region along this axis, is larger than the value obtained from (97) shown in Fig. 10 which corresponds to a cut of the domain [FORMULA] along the axis [FORMULA] or [FORMULA]. This is why the values obtained in (95) are higher than those one would derive from (97).

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Online publication: February 25, 2000
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