Gamma-Ray Bursts are precise distance indicators similar to Type Ia Supernovae?

We estimate the distance modulus to long gamma-ray bursts (LGRBs) using the Type I Fundamental Plane, a correlation between the spectral peak energy $E_{\rm p}$, the peak luminosity $L_{\rm p}$, and the luminosity time $T_{\rm L}$ ($\equiv E_{\rm iso}/L_{\rm p}$ where $E_{\rm iso}$ is isotropic energy) for small Absolute Deviation from Constant Luminosity(ADCL). The Type I Fundamental Plane of LGRBs is calibrated using 8 LGRBs with redshift $z<1.4$. To avoid any assumption on the cosmological model, we use the distance modulus of 557 Type Ia supernovae (SNeIa) from the Union 2 sample. This calibrated Type I Fundamental Plane is used to measure the distance moduli to 9 high-redshift LGRBs with the mean error $\bar σ_μ=0.31$, which is comparable with that of SNe Ia $\bar σ_μ=0.26$ where $μ$ stands for the distance modulus. The Type I Fundamental Plane is so tight that our distance moduli have very small uncertainties. From those distance moduli, we obtained the constraint $Ω_{\rm M}=0.22\pm0.04$ for flat $Λ$CDM universe. Adding 9 LGRBs distance moduli ($z>1.4$) to 557 SNeIa distance moduli ($z<1.4$) significantly improves the constraint for non-flat $Λ$CDM universe from ($Ω_{\rm M}, Ω_{\rm Λ}$)=($0.29\pm0.10$, $0.76\pm0.13$) for SNeIa only to ($Ω_{\rm M}, Ω_{\rm Λ}$)=($0.23\pm0.06$, $0.68\pm0.08$) for SNeIa and 9 LGRBs.

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