The Existence of an Old Quasar at $z = 3.91$ and its Implications for   $\Lambda(t)$ Deflationary Cosmologies
  We investigate some observational constraints on decaying vacuum cosmologies based on the recently discovered old high redshift quasar APM 08279+5255. This object is located at $z = 3.91$ and has an estimated age of 2-3 Gyr. The class of $\Lambda(t)$ cosmologies is characterized by a positive $\beta$ parameter smaller than unity which quantifies the ratio between the vacuum and the total energy density. Assuming the lower limit age (2 Gyr) and that the cold dark matter contributes with $\Omega_{\rm M}=0.2$ we show that $\beta$ is constrained to be $\ge 0.07$ while for an age of 3 Gyr and $\Omega_{\rm M}=0.4$ the $\beta$ parameter must be greater than 0.32. Our analysis includes closed, flat and hyperbolic scenarios, and it strongly suggests that there is no age crisis for this kind of $\Lambda(t)$ cosmologies. Lower limits to the redshift quasar formation are also briefly discussed to the flat case. For $\Omega_{\rm M}=0.4$ we found that the redshift formation is constrained by $z_{f}\ge 8.0$. 
J. V. Cunha (UFRN) R. C. Santos (UFRN)