4 Astron. Astrophys. 353, 63-71 (2000)
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Astron. Astrophys. 353, 63-71 (2000)

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2. Magnitude-redshift relation and homogeneous models

The luminosity distance [FORMULA] of a source is defined as the distance from which the radiating body, if motionless in an Euclidean space, would produce an energy flux equal to the one measured by the observer. It thus verifies


L being the absolute luminosity, i.e. the luminosity in the rest frame of the source, and l the measured bolometric flux, i.e. integrated over all frequencies by the observer.

In Friedmann-Lemaître-Robertson-Walker (FLRW) models, the distance measure at redshift z is a function of z and of the parameters of the model (Carroll et al. 1992)


[FORMULA] being the current Hubble constant, [FORMULA], the mass density parameter, and, [FORMULA] being defined as


where [FORMULA] is the cosmological constant and with


The apparent bolometric magnitude m of a standard candle of absolute bolometric magnitude M, at a given redshift z, is thus also a function of z and of the parameters of the model. Following Perlmutter et al. (1997), hereafter referred to as P97, it can be written, in units of megaparsecs, as


The magnitude "zero-point" [FORMULA] can be measured from the apparent magnitude and redshift of low-redshift examples of the standard candles, without knowing [FORMULA]. Furthermore, [FORMULA] depends on [FORMULA] and [FORMULA] with different functions of redshift. A priori assuming that FLRW models are valid to describe the observed universe, R98 and P99 thus proceed as follows to determine [FORMULA] and [FORMULA].

A set of apparent magnitude and redshift measurements for low-redshift ([FORMULA]) SNIa is used to calibrate Eq. (3), and another set of such measurements for high-redshift ([FORMULA]) is used to determine the best fit values of [FORMULA] and [FORMULA] 1. The results of these measurements are plotted in Figs. 4 and 5 of R98 and Figs. 1 and 2 of P99. The magnitude-redshift relation from the data and the theoretical curves obtained for different values of the cosmological parameters are then compared and best-fit confidence regions in the parameter space are plotted, see Figs. 6 and 7 of R98 and Fig. 7 of P99.

It is convenient at this stage to make the following remarks.

If one considers any cosmological model for which the luminosity distance [FORMULA] is a function of the redshift z and of the other cosmological parameters of the model, and if this function is Taylor expandable near the observer, i.e. around [FORMULA], the analysis of observational data at [FORMULA], in the framework of this model, can legitimately use the Taylor expansion


as, by definition of luminosity distances, [FORMULA].

Here, cp denotes the set of cosmological parameters, pertaining to the given model, which can be either constants, as in FLRW models, or functions of z, as in the example presented in Sects. 3 and 4.

Luminosity distance measurements of sources at different redshifts [FORMULA] yield values for the different coefficients in the above expansion. Going to higher redshifts amounts to measuring the coefficients of higher power of z. For very low redshifts, the leading term is first order; for intermediate redshifts, second order. Then third order terms provide significant contributions. For redshifts approaching unity, higher order terms can no longer be neglected.

Therefore, for cosmological models with very high (or infinite) number of free parameters, such data only provide constraints upon the values of the parameters near the observer. But for cosmological models with few parameters, giving independent contributions to each coefficient in the expansion, the method not only provides a way to evaluate the parameters, but, in most cases, to test the validity of the model itself.

For FLRW models, precisely, one obtains from Eq. (2)


These coefficients are independent functions of the three parameters of this class of models, [FORMULA] and [FORMULA]. With the method retained by P99 and described above, [FORMULA] is hidden in the magnitude "zero-point" [FORMULA]. The coefficients [FORMULA] of the expansion of [FORMULA] are thus functions of [FORMULA] and [FORMULA] alone. However, the same following remarks, with D replacing [FORMULA], apply to the analysis and results of R98.

In standard models, i.e. for [FORMULA]


If the analysis of the measurements gives [FORMULA], it implies [FORMULA], which is physically irrelevant. The model is thus ruled out. This is what happens with the SNIa data, and what induces R98 and P99 to postulate a strictly positive cosmological constant, to counteract the [FORMULA] term.

To test FLRW models with [FORMULA], one has to go at least to the third order to have a chance to obtain a result. This seems to be the order currently reached by the SNIa surveys. This last assertion rests on the two following remarks:

  1. As stressed in P97, R98 and P99, the well-constrained linear combination of [FORMULA] and [FORMULA] obtained from the data is not parallel to any contour of constant current "deceleration" 2 parameter [FORMULA]. As [FORMULA], this implies that higher order terms effectively contribute.

  2. The contribution of the fourth and higher order terms is negligeable. One can easily verify that, for the higher redshifts reached by the surveys, the contribution of the fourth order term does not overcome the measurement uncertainties, of the order of 5 to 10% (see Figs. 4 and 5 of R98 and Figs. 1 and 2 of P99).

A ruling out of FLRW models with non zero cosmological constant, at the third order level, i.e. due to a negative value for [FORMULA], would occur provided (see Eqs. (5) to (7))


This cannot be excluded by the results of R98 and P99. It corresponds to the left part of the truncated best-fit confidence ellipsoidal regions in the [FORMULA] plane of R98 Figs. 6 and 7 and P99 Fig. 7, and to the upper part of the error bars for the higher redshifts data in R98 Figs. 4 and 5 and P99 Figs. 1 and 2.

P99 propose an approximation of their results, which they write as


which corresponds, in fact, to:


It must be here stressed that the results published as primary by P99 and R98, under the form of best-fit confidence regions in the [FORMULA] plane, proceed from a Bayesian data analysis, for which a prior probability distribution accounting for the physically (in Friedmann cosmology) allowed part of parameter space is assumed. Results as given in the form of Eq. (11) are thus distorted by an a priori homogeneity assumption, which would have to be discarded for the completion of the test here proposed.

It may however occur that future, more accurate measurements yield values for the [FORMULA]s verifying Eq. (10) with the [FORMULA] sign, which would correspond to a physically consistent positive [FORMULA]. In this case, the FLRW models will have to be tested to the fourth order, i.e. with sources at redshifts nearer [FORMULA]. A final test for the homogeneity hypothesis on our past light cone would be a check of the necessary condition, obtained from Eqs. (5) to (8), and which can be written as


The above described method only applies to data issued from [FORMULA] supernovae. If the ongoing surveys were to discover more distant sources, up to redshifts higher than unity, the Taylor expansion would no longer be valid.

In practice, one will have to consider the Hubble diagram for the largest sample of accurately measured standard candles at every available scale of redshift from the lowest to the highest (e.g. Figs. 4 and 5 of R98 and Figs. 1 and 2 of P99). The FLRW theoretical diagram best fitting the data at intermediate redshift ([FORMULA] for measurements with 5% accuracy, [FORMULA] for 10% accuracy) will be retained as candidate. If it corresponds to a negative value for [FORMULA], the assumption of large scale homogeneity for the observed part of the universe will have to be discarded, and no higher redshift data will be needed, at least to deal with this issue. If it corresponds to a positive [FORMULA], the diagram will have to be extended to higher redshift data. If these data confirm the best fit of this candidate model up to [FORMULA] (for measurements at some 5-10% accuracy), the homogeneity hypothesis (and the corresponding values of the model parameters) would receive a robust support (provided no unlikely fine tuning of another model of universe parameters).

Another practical way of analysing the data is given by the very clever method described by Goobar and Perlmutter (1995), hereafter refered to as GP. An application of this method to probe large scale homogeneity of the observed universe would, in principle, imply the measurement of the apparent magnitude and redshift of only three intermediate and high redshift standard candles. In practice, more would certainly be needed to smooth out observational uncertainties. The method applies to sources with any redshift values and runs as follows.

Using Eqs. (2) and (3), one can predict the apparent magnitude of a source measured at a given redshift, in a peculiar FLRW model (i.e. for a given pair of values for [FORMULA] and [FORMULA]). Fig. 1 of GP shows the contours of constant apparent magnitude on the [FORMULA]-versus-[FORMULA] plane for two sufficiently different redshifts. When an actual apparent measurement of a source at a given redshift is made, the candidate FLRW model selected is the one with values for [FORMULA] and [FORMULA] narrowed to a single contour line. Since one can assume some uncertainty in the measurements, the allowed ranges of [FORMULA] and [FORMULA] are given by a strip between two contour lines. Two such measurements for sources at different redshift can define two strips that cross in a more narrowly constrained "allowed" region, shown as a dashed rhombus in Fig. 1 of GP. Now, if a third measurement of a source at a redshift sufficiently different from the two others is made, a third strip between two contour lines is selected. If this strip clearly crosses the previously drawn rhombus (and if the measurement uncertainties are kept within acceptable values), it provides support for the homogeneity hypothesis, with the above pointed out reserve. In case this strip clearly misses the rhombus, this hypothesis is ruled out, at least on our past light cone.

If the latter happens to be the case, we shall need an alternative model to fit the data. An example of a model, able to fulfil this purpose without the help of a cosmological constant is proposed in the following section.

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Online publication: December 8, 1999