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

## 4. Illustration: Flat LTB () models

To 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 R is given above by Eq. (19) and the calculation of the successive derivatives of R, contributing to the expressions for the expansion coefficients, is straightforward.

As the mass function remains constant with time, it can be used to define a radial coordinate r: , where is a constant.

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" universe 5. A decreasing has thus an analogous effect in a LTB universe to a positive cosmological constant in a FLRW one. They both make the observed universe look "older".

© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999