Cosmic web anisotropy is the primary indicator of halo assembly bias

The internal properties of dark matter haloes correlate with the large-scale halo clustering strength at fixed halo mass $-$ an effect known as assembly bias $-$ and are also strongly affected by the local, non-linear cosmic web. Characterising a halo’s local web environment by its tidal anisotropy $\alpha$ at scales $\sim 4$ x the halo radius, we demonstrate that these multi-scale correlations represent two distinct statistical links: one between the internal property and $\alpha$, and the other between $\alpha$ and large-scale ( $>30h^{-1}$Mpc) halo bias $b_1$. We focus on scalar internal properties of haloes related to formation time (concentration $c_{\mathrm{v}\mathrm{i}\mathrm{r}}$), shape (mass ellipsoid asphericity $c\mathrm{/}a$), velocity dispersion structure (velocity ellipsoid asphericity $c_v\mathrm{/}{a}_v$ and velocity anisotropy $\beta$) and angular momentum (dimensionless spin $\lambda$) in the mass range $8\times10^{11}< M_{\rm vir}/(h^{-1}M_\odot)<5\times10^{14}$. Using conditional correlation coefficients and other detailed tests, we show that the joint distribution of $\alpha$, $b_1$ and any of the internal properties $c\in \{\beta ,c_v\mathrm{/}{a}_v,c\mathrm{/}a,c_{\mathrm{v}\mathrm{i}\mathrm{r}},\lambda \}$ is consistent with $p(\alpha ,b_1,c)\simeq p(\alpha )p(b_1\mathrm{\mid }\alpha )p(c\mathrm{\mid }\alpha )$, at all but the largest masses. \textit{Thus, the assembly bias trendsc-b_1$ reflect the two fundamental correlations $c-\alpha$ and $b_1-\alpha$.}$ Our results are unaffected by the exclusion of haloes with recent major merger events or splashback objects, although the latter are distinguished by the fact that $\alpha$ does not explain their assembly bias trends. The overarching importance of $\alpha$ provides a new perspective on the nature of assembly bias of distinct haloes, with potential ramifications for incorporating realistic assembly bias effects into mock catalogs of future large-scale structure surveys and for detecting galaxy assembly bias.