## 4. Illustration: Flat LTB () modelsTo illustrate which kind of constraints can be imposed on LTB parameters by current observational results, the peculiar case of spatially flat LTB () models is analysed here. Spatial flatness is a property of the subclass of LTB models
verifying (Bondi 1947). In this
case, the expression for As the mass function remains
constant with time, it can be used to define a radial coordinate
With the covariant definition for above mentioned (Eq. (41)), the s, as derived from Eqs. (38) to (40), can thus be written, in units , in the form with the previously indicated choice , where is the time-like coordinate at the observer. It is convenient to note that is not a free parameter of the model, since its value proceeds from the currently measured temperature at 2.73 K (Célérier & Schneider 1998). A comparison with the corresponding FLRW coefficients gives the following relations: The above Eq. (47) implies that a non vanishing cosmological constant in a FLRW interpretation of data at corresponds to a mere constraint on the model parameters in a flat LTB () interpretation. Any magnitude-redshift relation, established up to the redshifts and with the precisions achieved by current measurements, i.e. at third order level, can thus be interpreted in either model. For instance, the latest results published in P99, and given under the form of Eq. (11), correspond, in a flat LTB () interpretation, to Such a result would imply a negative value for at least one of the
two quantities or
, which would be an interesting
constraint on the "Big-Bang" function in the observer's neighbourhood.
For instance, a function decreasing
near the observer would imply, for a source at a given
, an elapsed time from the initial
singularity that is longer in an LTB model than in the corresponding
FLRW one, i.e. an "older"
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