 Scalar expansion perturbation

With negative extrinsic curvature convention, the background trace is $\bar{K}=-3H$. The scalar expansion perturbation is $\kappa =K+3H$. For lapse function $N=\bar{N}(1+\Psi )$, induced metric $h_{ij}=a^2[(1-2\Phi ){\delta }_{ij}+2E_{,ij}]$ and shift vector $N_i=-a^2{B}_{,i}$, it is $\kappa =3(\dot{\Phi }/\bar{N}+H\Psi )-\Delta \chi$, where $\chi$ is the scalar shear potential. The expansion of the future normal congruence is $-K$, so its perturbation has the opposite sign.