Appendix B: Pullback and Lie-derivative identities

The differential-form construction in Chapter 2 uses only the standard functorial properties of pullbacks and Cartan’s formula. They are listed here to make clear which steps are geometric identities and which steps use cosmological dynamics.

For a smooth map \(F:N\to M\) and a \(p\)-form \(\alpha\) on \(M\), the pullback is defined by

\[ (F^*\alpha)_q(v_1,\ldots,v_p) =\alpha_{F(q)}(F_*v_1,\ldots,F_*v_p). \tag{B.1}\]

It satisfies

\[ \begin{aligned} F^*(\alpha\wedge\beta)&=F^*\alpha\wedge F^*\beta,\\ F^*(\mathrm d\alpha)&=\mathrm d(F^*\alpha),\\ (F\circ G)^*&=G^*F^*. \end{aligned} \tag{B.2}\]

For a vector field \(X\) with flow \(\varphi_\lambda\),

\[ \mathcal L_X\alpha =\left.\frac{\mathrm d}{\mathrm d\lambda}\varphi_\lambda^*\alpha \right|_{\lambda=0} =\mathrm d(\iota_X\alpha)+\iota_X\mathrm d\alpha. \tag{B.3}\]

These identities are sufficient to derive both current conservation and the linear pullback formula.

Two nowhere-vanishing top forms on the same oriented three-manifold differ by a scalar function. Thus the definition

\[ \mathsf X^*\boldsymbol{\mathcal J}=n_g^{\mathrm{obs}}\widetilde{\boldsymbol{\omega}} \tag{B.4}\]

is coordinate-independent. The scalar \(n_g^{\mathrm{obs}}\) is the Radon–Nikodym-type ratio of the physical count form to the fiducial observed-coordinate volume form.