Thus, although the pressure is not zero at the center of symmetry, r = 0, the effect of acceleration is nondetectable at the center, since the acceleration scalar vanishes there. This limit gives a value for the acceleration scalar u˙ of order |u˙|<~0.66×10 -10r Mpc -1, where r is the radial coordinate in the model. The choice of model parameters is quite robust: in order to obtain a good fit to the current data, one requires only that the non-uniform pressure parameter a in one of the models be negative and satisfy |a| <~ 3 km 2 s -2 Mpc -1. The Stephani models considered here could, therefore, significantly alleviate the conflict between recent cosmological and astrophysical age predictions. Our Stephani models, on the other hand, can give a good fit to the data with an age of up to 15 Gyr. Based on this value for H 0 and assuming Λ >= 0, the data would imply a Friedmann age of at most 13 Gyr and in fact a best-fit (for q 0 = 0.5) age of only 10 Gyr. Several recent calibrations of the Hubble parameter from the Hubble diagram of SN Ia's and other distance indicators indicate a value of H 0 ~= 65 and a Hubble time of ~15 Gyr. We claim that this paper is the first attempt to compare inhomogeneous models of the universe with real astronomical data.
We show that this age increase is obtained for a wide range of the non-uniform pressure parameters of the Stephani models. By a particular choice of model parameters, we show that these models can give an excellent fit to the observed redshifts and (corrected) B-band apparent magnitudes of the data, but for an age of the universe that is typically about 2 Gyr-and may be more than 3 Gyr-greater than in the corresponding Friedmann model, for which nonnegative values of the deceleration parameter appear to be favored by the data. We use the redshift-magnitude relation, as derived by Dąbrowski, for the two exact non-uniform pressure spherically symmetric Stephani universes with the observer positioned at the center of symmetry in order to test the agreement of these models with recent observations of high-redshift Type Ia supernovae (SN Ia's).
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