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Write $R(v)={\int }_{\mathbb{R}}v(u{)}^{2}du$. The kernel density estimator with bandwidth $h>0$ isSufficient conditions are that $K$ be a nonnegative symmetric probability density function with $R(K)<\infty$ and finite nonzero second moment ${\mu }_2(K)=\int u^{2}K(u)du$, and that $h\to 0$, $nh\to \infty$. Symmetry and the second-moment assumption give $\int uK(u)du=0$. The expectation is a convolution:Taylor's theorem with integral remainder givesThe zeroth-order term integrates to $f(x)$, and the first-order term vanishes. The second derivative is bounded and continuous, so dominated convergence, with dominating function proportional to $u^{2}K(u)$, yieldsCombining this with the supplied leading variance gives the asymptotic mean squared errorWhen $f(x)>0$ and $f^{\prime \prime }(x)\ne 0$, its derivative is $-R(K)f(x)/(n{h}^2)+{\mu }_2(K{)}^2{f}^{\prime \prime }(x{)}^2{h}^3$. It changes sign exactly once, giving the pointwise optimal kernel bandwidthPositivity of $f(x)$ is needed for this positive interior optimum. The printed condition $f^{\prime \prime }(x)\ne 0$ alone does not ensure it. For instance, $f(x)=x^2\varphi (x)$ is a normalized bounded probability density function with bounded continuous square-integrable second derivative, but $f(0)=0$ and $f^{\prime \prime }(0)=2\varphi (0)>0$. At this point the displayed leading variance vanishes and the two-term AMSE has no positive interior minimizer; finer variance terms are needed to determine an appropriate bandwidth. The formal formula there gives zero, which does not satisfy the bandwidth assumptions.

For integrated error, the same expansion holds in $L^2$: translation continuity of $f^{\prime \prime }\in L^2$ applied to the integral remainder givesIndeed Minkowski's integral inequality bounds the norm of the difference, divided by $h^2$, by the integral of $u^{2}K(u){\int }_0^1(1-s)\Vert f^{\prime \prime }(\cdot -shu)-f^{\prime \prime }{\Vert }_{2}dsdu$; each translated difference tends to zero and is bounded by $2\Vert f^{\prime \prime }{\Vert }_2$. This proves the claimed integrated bias expansion without assuming a uniform pointwise remainder over the whole real line.

If $K_h(x)=h^{-1}K(x/h)$, direct integration of the variance givesThe last term is $O(n^{-1})$ because $f$ is bounded and integrable, hence square integrable, and convolution with a probability density function does not increase its $L^2$ norm. It is negligible compared with $1/(nh)$. Thus the asymptotic mean integrated squared error and its minimizer areHere $R(f^{\prime \prime })>0$: otherwise the continuous second derivative vanishes everywhere, making $f$ affine, impossible for a probability density function on the whole real line.

Now take the standard normal density $f=\varphi$. Differentiation gives ${\varphi }^{\prime \prime }(x)=(x^2-1)\varphi (x)$. Gaussian integration givesThe kernel constants cancel from the bandwidth ratio. At $x\ne \pm 1$,Put $t=x^2$. The logarithmic derivative of $e^{t/2}/(t-1{)}^2$ is $1/2-2/(t-1)$. On $0\le t<1$ the function increases, giving its minimum at $t=0$, with value one. On $t>1$ it decreases up to $t=5$ and then increases, giving its minimum $e^{5/2}/16<1$. Thus the global minimizers are $x=\pm \sqrt{5}$, and the normal-density local-to-global bandwidth ratio isThe ratio diverges at the two inflection points $\pm 1$, where the second-order pointwise bias approximation vanishes and needs a higher-order replacement.

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.

For nonzero momentum, the Fourier transform of the local bias expansion isThe subtracted variance contributes only at $\mathbf{k}=0$. Because $\delta$ is a Gaussian random field, its three-point function vanishes and its four-point function factorizes by Wick theorem. At leading order one of the three galaxy fields supplies the quadratic term and the other two supply linear terms. The two cross-contractions cancel the factor $1/2$, givingThus even Gaussian matter fluctuations acquire a nonzero galaxy bispectrum through local quadratic galaxy bias.

In standard perturbation theory in cosmology, ${\delta }^{(n)}$ is an $n$-leg vertex carrying the symmetrized standard perturbation theory density kernel $F_n$. For Gaussian random fields, a connected four-point diagram with $L$ loops contains $3+L$ linear power spectra, hence $6+2L$ linear fields.

At tree level the required perturbative-order partitions areThe first is a star with one $F_3$ vertex and three linear external legs. The second has two $F_2$ vertices joined by one internal power-spectrum line, with one linear external leg attached to each vertex. At one loop all five partitions of eight into four positive parts are required:These are conventionally denoted $T_{5111}$, $T_{4211}$, $T_{3311}$, $T_{3221}$, and $T_{2222}$; all inequivalent placements of external legs and all Wick contractions are understood. This list is the complete set of one-loop matter trispectrum topologies.

Let $P_a=P_L(k_a)$ and ${\mathbf{k}}_{ab}={\mathbf{k}}_a+{\mathbf{k}}_b$. With ${\sum }_a{\mathbf{k}}_a=0$, the star contribution iswhere $\{a,b,c\}=\{1,2,3,4\}\setminus \{d\}$. For the exchange contribution, sum over the six unordered choices $\{a,b\}$ of linear external legs and let $\{c,d\}$ be the complementary pair:Thus the connected tree trispectrum is $T_{\mathrm{tree}}=T_{3111}+T_{2211}$.

The pressureless single-stream equations cease to be a complete one-loop prediction because loop momenta probe short nonlinear scales and produce ultraviolet-sensitive terms. A consistent result must include the effective field theory of large-scale structure: effective-stress counterterms, including their insertions into the tree topologies, and stochastic terms at the order allowed by mass conservation and momentum conservation. Their coefficients absorb short-scale dependence and must be fitted or matched. If the observable is a biased tracer rather than matter itself, the corresponding renormalized bias expansion is also required.