$H_0$ gets cancelled, hence it does not introduce any uncertainty to the results.

![ The cumulative probability distributions with the matter parameters $\Omega_m=0.285$ (blue), $\Omega_m=0.300$ (black) and $\Omega_m=0.315$ (red) in $\Lambda$CDM cosmology.](fig5.eps){width="0.9\linewidth"}

![ The corresponding cumulative probability distributions with the matter parameters $\Omega_m=0.260$ (blue), $\Omega_m=0.275$ (yellow) and $\Omega_m=0.305 $(red) in DGP cosmology.](fig6.eps){width="0.9\linewidth"}

   Cosmological model  Cosmological parameter   Probability
  -------------------- ------------------------ -------------
      $\Lambda$CDM     $\Omega_m=0.285$         $10.00\%$
                       $\Omega_m=0.300$         $99.99\%$
                       $\Omega_m=0.315$         $10.00\%$
          DGP          $\Omega_m=0.260$         $10.00\%$
                       $\Omega_m=0.275 $        $99.99\%$
                       $\Omega_m=0.305 $        $10.00\%$

  : Summary of the cosmological constraints using strong gravitational lenses with Gaussian Processes.

We start our analysis with the $\Lambda$CDM model with constant dark energy density and constant cosmic equation of state $w=-1$. The corresponding cumulative probability distributions are plotted in Fig. 5, which also locate the $\Delta A$ values with different matter density parameter: $\Omega_m=0.300$ (black), $\Omega_m=0.285$ (blue), and $\Omega_m=0.315$ (red). The probabilities associated with these differential areas are summarized in Table 1. As can be clearly seen from the results, the probability of the matter density parameter $\Omega_m=0.300$ being consistent with the GP reconstructed $\mathcal{D}_{\rm obs}$ function is $99.99\%$, while the probabilities that the matter density parameter $\Omega_m=0.285$ and $\Omega_m=0.315$ being inconsistent with the current SGL observations are $90\%$. Considering the additional 