Linear Lagrangian halo bias ($b_L(M)$)

Linear Lagrangian halo bias is the fractional response of halo abundance to a long-wavelength overdensity in the initial coordinates. For Press-Schechter halos, $b_L=({\nu }^2-1)/{\delta }_c$.

Mapping matter and halos from their initial positions adds the matter displacement contribution, giving $b=1+b_L$. Mass conservation requires the mass-weighted consistency relation $\int (dn/dM)(M/\bar{\rho })b(M)dM=1$.