  Halo assembly bias is the secondary dependence of the clustering of
dark-matter haloes on their assembly histories at fixed halo mass. This
established dependence is expected to manifest itself on the clustering of the
galaxy population, a potential effect commonly known as galaxy assembly bias.
Using the IllustrisTNG300 magnetohydrodynamical simulation, we analyse the
dependence of the properties and clustering of galaxies on the shape of the
specific mass accretion history of their hosting haloes (sMAH). We first show
that several halo and galaxy properties strongly correlate with the slope of
the sMAH ($\beta$) at fixed halo mass. Namely, haloes with increasingly steeper
$\beta$ increment their halo masses faster at early times, and their hosted
galaxies present larger stellar-to-halo mass ratios, lose their gas faster,
reach the peak of their star formation histories at higher redshift, and become
quenched earlier. We also demonstrate that $\beta$ is more directly connected
to these key galaxy formation properties than other broadly employed halo
proxies, such as formation time. Finally, we measure the secondary dependence
of galaxy clustering on $\beta$ at fixed halo mass as a function of redshift.
By tracing back the evolution of individual haloes, we show that the amplitude
of the galaxy assembly bias signal for the progenitors of $z=0$ galaxies
increases with redshift, reaching a factor of 2 at $z = 1$ for haloes of
$M_\mathrm{halo}=10^{11.5}-10^{12}$ $h^{-1}\mathrm{M}_\odot$. The measurement
of the evolution of assembly bias along the merger tree provides a new
theoretical perspective to the study of secondary bias. Our findings, which
show a tight relationship between halo accretion and both the clustering and
the observational properties of the galaxy population, have also important
implications for the generation of mock catalogues for upcoming cosmological
surveys.
