Scale-dependent halo bias from local non-Gaussianity ($\Delta b(k)=2f_{\mathrm{NL}}{\delta }_c{b}_L/\alpha (k)$)

For a universal abundance depending on halo peak height, the peak-background split gives a potential-response coefficient $b_{\phi }=2f_{\mathrm{NL}}{\delta }_c{b}_L$, with $b_L$ the linear Lagrangian halo bias. If $\delta =\alpha (k)\phi$, the corresponding deterministic galaxy bias correction is $\Delta b=b_{\phi }/\alpha (k)$, proportional to $k^{-2}$ on sufficiently large scales. For a physical Eulerian tracer, $b_E=1+b_L$, so $b_{\phi }=2f_{\mathrm{NL}}{\delta }_c(b_E-1)$. The deterministic tracer power spectrum is $(b_E+\Delta b{)}^2{P}_m$; it remains nonnegative even if a negative correction changes the sign of the tracer-matter correlation.