The age problem in $\Lambda$CDM model

The age problem in the $\Lambda$CDM model is re-examined. We define the elapsed time $T$ of an object is its age plus the age of the Universe when it was born. Therefore in any cosmology, $T$ must be smaller than the age of the Universe. For the old quasar APM 08279+5255 at $z=3.91$, previous studies have determined the best-fit value of $T$, 1 $\sigma$ lower limit and the lowest limit to $T$ are 2.3, 2.0 and 1.7 Gyr, respectively. Constrained from SNIa$+R+A+d$, SNIa$+R+A+d+H(z)$, and WMAP5+2dF+SNLS+HST+BBN, the $\Lambda$CDM model can only accommodate $T(z=3.91)=1.7$ Gyr at 1 $\sigma$ deviation. Constrained from WMAP5 results only, the $\Lambda$CDM model can only accommodate $T(z=3.91)=1.7$ Gyr at 2 $\sigma$ deviation. In all these cases, we found that $\Lambda$CDM model accommodates the total age (14 Gyr for $z=0$) of the Universe estimated from old globular clusters, but cannot accommodate statistically the 1 $\sigma$ lower limit to the best-fit age of APM~08279+5255 at $z=3.91$. These results imply that the $\Lambda$CDM model may suffer from an age problem.