Utilizing the Updated Gamma-Ray Bursts and Type Ia Supernovae to Constrain the Cardassian Expansion Model and Dark Energy

We update gamma-ray burst (GRB) luminosity relations among certain spectral and light-curve features with 139 GRBs. The distance modulus of 82 GRBs at $z>1.4$ can be calibrated with the sample at $z\le 1.4$ by using the cubic spline interpolation method from the Union2.1 Type Ia supernovae (SNe Ia) set. We investigate the joint constraints on the Cardassian expansion model and dark energy with 580 Union2.1 SNe Ia sample ($z<1.4$) and 82 calibrated GRBs data ($1.4<z\le 8.2$). In $\mathrm{\Lambda }$CDM, we find that adding 82 high-\emph{z} GRBs to 580 SNe Ia significantly improves the constrain on ${\mathrm{\Omega }}_m-{\mathrm{\Omega }}_{\mathrm{\Lambda }}$ plane. In the Cardassian expansion model, the best fit is ${\mathrm{\Omega }}_m=0.24_{-0.15}^{+0.15}$ and $n=0.16_{-0.52}^{+0.30}$ $(1\sigma )$, which is consistent with the $\mathrm{\Lambda }$CDM cosmology $(n=0)$ in the $1\sigma$ confidence region. We also discuss two dark energy models in which the equation of state $w(z)$ is parametrized as $w(z)=w_0$ and $w(z)=w_0+w_{1}z\mathrm{/}(1+z)$, respectively. Based on our analysis, we see that our Universe at higher redshift up to $z=8.2$ is consistent with the concordance model within $1\sigma$ confidence level.