## 6. Derivation of cosmological parametersIn practice, one uses the Type Ia supernovae as standard candles in
order to derive the cosmological parameters
and
from the observed relation redshift
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 where is the observed distance
defined in (18) seen as a function of
and
which are thus determined by the
observation (if there is no distortion by weak lensing:
, these parameters are equal to the
actual cosmological parameters and
). 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
obeys:
. Thus, for a given redshift
of the source and a peculiar
cosmology of the actual universe we
can obtain "confidence regions" in the
plane for the observed parameters
and
. This means that for any
with
, the "observed universe" has a
probability to lie within the
domain . Thus, for any point
we calculate through (94) the weak
lensing magnification We display in Fig. 9 these "confidence regions"
we obtain for the three cosmologies
we have studied in details in the previous sections, for the redshifts
and .
At a given redshift , we show the
domains defined by
(dashed region within two solid
lines) and by (within the two
dashed lines). This means that from one observation at a given
redshift the observer will conclude with a probability of
that the cosmological parameters
lie in the dashed domain. We also
display the boundary line given by
within (94) (solid line on the left
side). Thus, because of the lower bound
the observed parameters
cannot lie to the left of the line
(of course, here we only consider
the effects of weak lensing). The cross is the point
. For a given redshift, all these
boundary lines are parallel. Moreover, their slope (when seen as a
function of
) increases with
. Indeed, as emphasized by Perlmutter
et al.(1997) the parameters and
enter
with different powers of
so that the observed distance
is not a function of
(except in the limit
) but of a combination of
and
which varies with
. This implies the drift with
of the slope of the boundary lines
we defined above. Of course, this is the reason why observations can
simultaneously constrain and
(provided one has a finite range of
source redshifts). The sign of the slope of these strips translates
the fact that for the same and
redshift a flat universe corresponds to a larger distance
than an open geometry. Of course,
the strong asymmetry of the probability distribution
, and of the intervals
, leads to asymmetric regions
. Thus, as
increases the strip
grows but it is bounded on the left
by while it can extend to infinity
to the right. Indeed, as can be seen from (94) larger Note that if both observations (at and ) are independent the intersection of the regions labelled , used in (95), corresponds to a probability . 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 roughly decreases as ). Moreover, we can check in the figure that observations can unambiguously discriminate between and . In the case of a low-density universe one can also clearly discriminate between and .
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 of the sources, we show in Fig. 10 the quantity:
Here is the absolute magnitude of the supernova and is the magnitude dispersion due to weak lensing. Thus, we have: where is obtained from (94). For
low-density universe we eliminate
in (94) by assuming the right cosmology (flat or
) so that
is a function of
al one. In the case of a critical
universe we display the results we obtain when we assume
or
. We use for all redshifts
mag for the
lightcurve-width-corrected luminosity dispersion of SNeIa, see
Perlmutter et al.(1999). The solid lines in Fig. 10 show
from (97). The dashed curves show
the inaccuracy on due to
alone while the dotted curves show
the effect o alone. We can see that
for low-density universes the error due to the intrinsic dispersion
dominates up to
while for a critical universe the
weak lensing contribution already dominates for
. Note also that the inaccuracy of
the measure of is much larger for a
critical universe, which has a non-negligible probability to appear as
an open universe with or a flat
universe with for
. In particular, note that going to
high redshift does not increase the
accuracy of the determination of by
much. Thus, the minimum dispersion of
is
for low-density universe and
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
© European Southern Observatory (ESO) 2000 Online publication: February 25, 2000 |