their work for proofs and more details). Let $\mathcal{P}^-_x$ be the joint law of a cell system $(\chi_u:u\in\mathbb{U})$ starting from an Eve cell $\chi_\emptyset$ with initial size $x>0$ and a distinguished leaf $\widehat{\ell}\in\partial\mathbb{U}$, such that the law of $(\chi_u:u\in\mathbb{U})$ under $\mathcal{P}_x^-$ is absolutely continuous with respect to its law under $\mathcal{P}_x$, with density $x^{-\omega_-}\mathcal{M}$, and $\widehat{\ell}$ has conditional law $\mathcal{A}(\cdot)/\mathcal{M}$ given the cell system.

The *spine* is the process following in time the size of the ancestor of $\widehat{\ell}$. Up to modifying the genealogy, we can consider the spine to be the Eve cell of the cell-system, the law of the growth-fragmentation remains unchanged (see [@BBCK16] Section 4.2). Whereas the spine has a tilted distribution, its daughters generate, given the spine, independent growth-fragmentations with laws $(\mathcal{P}_x)_{x>0}$ according to their initial sizes. Its law is that of a positive self-similar Markov process $Y^-=(Y^-(t))_{t\geq 0}$ with same index $\alpha$. Let $\mathbb{P}_x^-$ be the law of $Y^-$ starting from $x>0$. As for $X$, we have that $(Y^-(t))_{t\geq 0}=(\exp(\eta^-(\tau_t)))_{t\geq 0}$, where $\eta^-$ is a Lévy process and $(\tau_t)_{t\geq 0}$ the associated time-change from the Lamperti’s transformation. (It will always be clear from the context whether $\tau_t$ denotes the time-change associated with $Y^-$ or $X$.) 