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Cosmological observables could be used to construct cosmological models,however, a fixed number of observables limited on the light cone is not enoughto uniquely determine a certain model. A reconstructed spherically symmetric,inhomogeneous model that share the same angular-diameter-distance-redshiftrelationship $d_A(z)$ and Hubble parameter $H(z)$ besides $\Lambda\textrm{CDM}$model (which we call LTB-$\Lambda\textrm{CDM}$ model in this paper), mayprovide another solution. Cosmic age, which is off the light cone, could beemployed to distinguish these two models. We derive the formulae for agecalculation with origin conditions. From the data given by 9-year WMAPmeasurement, we compute the likelihood of the parameters in these two modelsrespectively by using the Distance Prior method and do likelihood analysis bygenerating Monte Carlo Markov Chain for the purpose of breaking the degeneracyof $\Omega_m$ and $H_0$ (the parameters that we use for calculation). Theresults yield to be: $t_{\Lambda\textrm{CDM}} =13.76 \pm 0.09 ~\rm Gyr$,$t_{\rm {LTB}-\Lambda\textrm{CDM}} =11.38 \pm 0.15 ~\rm Gyr$, both in $1\sigma$agreement with the constraint of cosmic age given by metal-deficient stars. Thecosmic age method that is set in this paper enables us to distinguish betweenthe inhomogeneous model and $\Lambda\textrm{CDM}$ model.

Distinguishing between inhomogeneous model and $\Lambda\textrm{CDM}$ model with the cosmic age method