Metallicity evolution of direct collapse black hole hosts: CR7 as a case study

In this study we focus on the $z\sim6.6$ Lyman-$\alpha$ CR7 consisting of clump A that is host to a potential direct collapse black hole (DCBH), and two metal enriched star forming clumps B and C. In contrast to claims that signatures of metals rule out the existence of DCBHs, we show that metal pollution of A from star forming clumps clumps B and C is inevitable, and that A can form a DCBH well before its metallicity exceeds the critical threshold of $10^{-5}-10^{-6}\ \rm Z_{\odot}$. Assuming metal mixing happens instantaneously, we derive the metallicity of A based on the star formation history of B and C. We find that treating a final accreting black hole of $10^6-10^7\ \rm M_{\odot}$ in A for nebular emission already pushes its $H_{160}$ - [3.6] and [3.6]-[4.5] colours into the 3$\sigma$ limit of observations. Hence, we show that the presence of metals in DCBH hosts is inevitable, and that it is the coevolution of the LW radiation field and metals originating from neighbouring galaxies that governs DCBH formation in a neighbouring {initially} pristine atomic cooling haloes.