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Evaluate all derivatives at zero long perturbation, where $\nu ={\delta }_c/{\sigma }_{G,s}$, and write $\bar{n}(\nu )$ for the unperturbed abundance. The peak-background split changes the halo peak height toThe chain rule in the specified bias expansion therefore givesNo explicit abundance law is specified, so these derivatives are the general answer. For example, if $\bar{n}$ has the Press-Schechter halo mass function dependence $\nu e^{-{\nu }^2/2}$, then $b_{10}=({\nu }^2-1)/{\delta }_c$ and $b_{01}=2f_{\mathrm{NL}}({\nu }^2-1)$.

For the linear long mode, the cosmological Poisson equation gives ${\phi }_{G,L}(\mathbf{k})={\delta }_{G,L}(\mathbf{k})/\alpha (k)$. Consequently the deterministic galaxy bias and power spectrum areHere $P_m$ is the linear cosmological density power spectrum; stochastic tracer noise has been omitted. This scale-dependent halo bias from local non-Gaussianity is proportional to $k^{-2}$ in the specified $\alpha (k)=2k^2/(3H^2{\Omega }_m)$ convention.

For $b_{10}\ne 0$, plot the ratio $P_g/(b_{10}^2{P}_m)$ to separate the effect from the shape of $P_m$. Positive $f_{\mathrm{NL}}$ increases the ratio as $k$ decreases. Negative $f_{\mathrm{NL}}$ first suppresses it, gives a zero when $\alpha (k)=2|f_{\mathrm{NL}}|{\delta }_c$, and then makes it rise again after the effective galaxy bias changes sign. The auto-power spectrum never becomes negative. A truncation $P_g\simeq b_{10}^2[1+4f_{\mathrm{NL}}{\delta }_c/\alpha (k)]P_m$ is valid only when $|2f_{\mathrm{NL}}{\delta }_c/\alpha (k)|\ll 1$ and must not be extrapolated through the zero. Within the deterministic expression, both signs give $P_g\propto k^{-4}{P}_m$ at sufficiently small $k$. For a nearly scale-invariant primordial spectrum, $P_m\propto k^{n_s}$ in the large-scale transfer limit, so this formal asymptote is $k^{n_s-4}$.

There is a convention issue in calling the specified abundance response an observed galaxy overdensity. The threshold response is naturally a linear Lagrangian halo bias. If $b_{10}$ instead denotes the physical linear Eulerian halo bias, the Lagrangian-to-Eulerian linear bias relation adds one to the density coefficient: $b_{01}=2f_{\mathrm{NL}}{\delta }_c(b_{10}-1)$, and $P_g=[b_{10}+b_{01}/\alpha {]}^2{P}_m$. The boxed formulas follow the abundance expansion explicitly supplied in the paper. The Newtonian treatment also cannot be extrapolated beyond its physical large-scale domain without relativistic projection effects.

On wavelengths much larger than halos, all matter is partitioned among halos. The mass-weighted halo overdensity must therefore equal the matter overdensity. Since ${\delta }_h(M)=b(M){\delta }_m$, mass conservation requires

For the Press-Schechter formalism, the mass-fraction measure becomesIt is normalized and is a half-normal distribution, soUsing the linear Eulerian halo biastherefore giveswhich verifies the consistency relation.