Constraints from the Old Quasar APM 08279+5255 on Two Classes of $\Lambda($t$)$-Cosmologies

The viability of two different classes of $\Lambda(t)$CDM cosmologies is tested by using the APM 08279+5255, an old quasar at redshift $z = 3.91$. In the first class of models, the cosmological term scales as $\Lambda(t)\sim R^{-n}$. The particular case $n=0$ describes the standard $\Lambda$CDM model whereas $n=2$ stands for the Chen and Wu model. For an estimated age of 2 Gyr, it is found that the power index has a lower limit $n> 0.21$, whereas for 3 Gyr the limit is $n > 0.6$. Since $n$ can not be so large as $\sim 0.81$, the $\Lambda$CDM and Chen and Wu models are also ruled out by this analysis. The second class of models is the one recently proposed by Wang and Meng which describes several $\Lambda(t)$CDM cosmologies discussed in the literature. By assuming that the true age is 2 Gyr it is found that the $\epsilon$ parameter satisfies the lower bound $\epsilon > 0.11$, while for 3 Gyr, a lower limit of $\epsilon > 0.52$ is obtained. Such limits are slightly modified when the baryonic component is included.