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 $\bar \sigma_{\mu}=0.31$, which is comparable with
that of SNe Ia $\bar \sigma_{\mu}=0.26$ where $\mu$ 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 $\Omega_{\rm M}=0.22\pm0.04$ for flat $\Lambda$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 $\Lambda$CDM universe from
($\Omega_{\rm M}, \Omega_{\rm \Lambda}$)=($0.29\pm0.10$, $0.76\pm0.13$) for
SNeIa only to ($\Omega_{\rm M}, \Omega_{\rm \Lambda}$)=($0.23\pm0.06$,
$0.68\pm0.08$) for SNeIa and 9 LGRBs.
