Table 9: Scaling relations for the distant clusters.
  $\Omega_{\rm M} = 0.3, \Omega_{\Lambda}=0.7$, $z_{\rm col} = z_{\rm obs}$
Relation $\Delta _{\rm c}=500$ $\Delta _{\rm c}=500\Omega ^{0.427}$ $\Delta_{\rm c}=18\pi^2\Omega^{0.427}$
$L_{\rm X,bol}-T$ $6.53^{+8.34}_{-3.82}\times10^{42} (kT)^{3.08^{+0.48}_{-0.45}}$ $7.94^{+8.92}_{-4.58}\times 10^{42}(kT)^{3.01^{+0.47}_{-0.42}}$ $1.07^{+1.26}_{-0.55}\times 10^{43}(kT)^{2.92^{+0.40}_{-0.43}}$
M-T $1.64^{+0.35}_{-0.26}\times10^{13} (kT)^{1.68^{+0.10}_{-0.11}}$ $2.07^{+0.35}_{-0.43}\times 10^{13}(kT)^{1.68^{+0.13}_{-0.08}}$ $3.31^{+0.73}_{-0.60}\times 10^{13}(kT)^{1.69^{+0.11}_{-0.11}}$
$M_{\rm gas}-T$ $2.88^{+0.93}_{-0.64}\times10^{12} (kT)^{1.85^{+0.14}_{-0.15}}$ $4.32^{+1.28}_{-0.89}\times 10^{12}(kT)^{1.80^{+0.13}_{-0.15}}$ $1.10^{+0.29}_{-0.25}\times 10^{13}(kT)^{1.67^{+0.15}_{-0.13}}$
$f_{\rm gas}-T$ 0.28+0.09-0.06(kT)-0.08+0.14-0.14 0.33+0.10-0.08(kT)-0.11+0.15-0.15 0.47+0.16-0.15(kT)-0.19+0.20-0.16
$E_z^{-1}L_{\rm X,bol}-T$ $5.50^{+7.92}_{-3.24}\times10^{42} (kT)^{3.12^{+0.51}_{-0.48}}$ $7.24^{+9.15}_{-4.04}\times 10^{42}(kT)^{3.00^{+0.45}_{-0.45}}$ $1.01^{+0.99}_{-0.55}\times 10^{43}(kT)^{2.89^{+0.43}_{-0.39}}$
EzM-T $1.73^{+0.36}_{-0.32}\times10^{13} (kT)^{1.71^{+0.11}_{-0.10}}$ $2.11^{+0.45}_{-0.37}\times 10^{13}(kT)^{1.73^{+0.11}_{-0.11}}$ $3.51^{+0.75}_{-0.62}\times 10^{13}(kT)^{1.71^{+0.11}_{-0.10}}$
$E_zM_{\rm gas}-T$ $3.05^{+0.94}_{-0.72}\times10^{12} (kT)^{1.88^{+0.15}_{-0.15}}$ $4.62^{+1.18}_{-1.04}\times 10^{12}(kT)^{1.82^{+0.15}_{-0.12}}$ $1.19^{+0.30}_{-0.28}\times 10^{13}(kT)^{1.68^{+0.15}_{-0.12}}$


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