Appendix A: Response conventions and functional derivatives

This appendix collects normalization choices that are easy to conflate in applications. A derivative of the abundance and a derivative of its logarithm agree at first order but differ at nonlinear order, and a functional response kernel should be distinguished from the coefficients of a chosen local operator basis.

The response coefficients can be defined using derivatives of either \(\bar n_g\) or \(\ln\bar n_g\). For a homogeneous density background \(\Delta\), define

\[ b_N =\frac{1}{\bar n_g} \left.\frac{\partial^N\bar n_g}{\partial\Delta^N}\right|_0, \qquad \beta_N =\left.\frac{\partial^N\ln\bar n_g}{\partial\Delta^N}\right|_0. \tag{A.1}\]

Then

\[ b_1=\beta_1, \qquad b_2=\beta_2+\beta_1^2, \qquad b_3=\beta_3+3\beta_1\beta_2+\beta_1^3. \tag{A.2}\]

The notes use the \(b_N\) convention because it is the coefficient in the Taylor expansion of the fractional abundance.

For a functional \(F[\mathcal O]\), the derivative is defined by

\[ \delta F =\int\mathrm d^4x\, \frac{\delta F}{\delta\mathcal O^A(x)} \delta\mathcal O^A(x). \tag{A.3}\]

The response kernel in Eq. 1.9 is a distribution. Spatial locality does not mean it is exactly proportional to a spatial Dirac delta. Rather, the kernel has support over a region of size \(R_*\) and admits the derivative expansion

\[ \int\mathrm d^3r\,W_R(\boldsymbol{r})\mathcal O(\boldsymbol{x}+\boldsymbol{r}) =\mathcal O(\boldsymbol{x}) +c_2R_*^2\nabla^2\mathcal O(\boldsymbol{x})+\cdots \tag{A.4}\]

for long modes. Eq. 1.12 keeps the leading term.

A scalar separate-universe response probes only isotropic backgrounds. Tidal response requires an anisotropic long-wavelength background, and tensor-valued response such as Eq. 5.9 requires retaining the free tensor indices before imposing rotational invariance.