We compare the amplitudes of fluctuations probed by the 2dF Galaxy Redshift Survey and by the latest measurements of the Cosmic Microwave Background anisotropies. By combining the 2dFGRS and CMB data we find the linear-theory rms mass fluctuations in 8 Mpc/h spheres to be sigma_8 = 0.73 +-0.05 (after marginalization over the matter density parameter Omega_m and three other free parameters). This normalization is lower than the COBE normalization and previous estimates from cluster abundance, but it is in agreement with some revised cluster abundance determinations. We also estimate the scale-independent bias parameter of present-epoch L_s = 1.9L_* APM-selected galaxies to be b(L_s,z=0) = 1.10 +- 0.08 on comoving scales of 0.02 < k < 0.15 h/Mpc. If luminosity segregation operates on these scales, L_* galaxies would be almost un-biased, b(L_*,z=0) = 0.96. These results are derived by assuming a flat Lambda-CDM Universe, and by marginalizing over other free parameters and fixing the spectral index n=1 and the optical depth due to reionization tau=0. We also study the best fit pair (Omega_m,b), and the robustness of the results to varying n and tau. Various modelling corrections can each change the resulting b by 5-15 per cent. The results are compared with other independent measurements from the 2dFGRS itself, and from the SDSS, cluster abundance and cosmic shear.
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