 cooling in additional low-mass mini-haloes and reduce the number of DCBH host candidates, while increasing the threshold could have the opposite effect below $z\sim 16$.  To address this caveat, we have re-run our models  using the fitting formulae from \citet{kulkarni20}. We found that the differences in our results were almost negligible at very high redshift, although below $z=15$, it resulted in a few percent higher number of candidates relative to our fiducial case. However, we caution that our ability to fully test this caveat is limited by the resolution of our simulations (with a particle mass of $M_{\rm dm}\approx 9\times 10^4\,{\rm M_\odot}$, we are unable to resolve mini-haloes with masses much below $10^6\,{\rm M_\odot}$).

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth,trim=0.5cm 0.5cm 2cm 1.5cm,clip]{MinMass.pdf}
    \caption{Minimum mass for efficient H$_2$ cooling from Eq.~\eqref{eq:mmin} for different values of $J_{21}$ (red lines), compared with the results by \citet{kulkarni20}, shown as blue lines, and the redshift-independent result (for $J_{21}=0$) by \citet{schauer19}.}
    \label{fig:mmin}
\end{figure}

Above this mass threshold, a plausible process that is able to counteract radiative cooling is the compressional heating provided by the mass accretion onto the halo. This has been dubbed "dynamical heating" in the literature, 
and the corresponding heating rate $\Gamma_{\rm dyn}$ can be approximated as 
\begin{equation}
    n\Gamma_{\rm dyn} = \frac{n_{\rm H}}{0.76\mu}\frac{k_{\rm B}}{\gamma-1}\left(\frac{2\alpha_{\rm vir}}{3}M_{\rm vir}^{-1/3}\frac{{\rm d} M_{\rm vir}}{{\rm d} t} + \frac{T_{\rm vir}}{1+z}\frac{{\rm d}z}{{\rm d}t}\right),
    \label{eq:dynheat}
\end{equation}
where the first term on the right-hand side represents compressional heating accompanying the mass growth of the halo \citep{yoshida03}, and the second term accommodates the decrease of the characteristic densities and the virial temperature (at fixed halo mass) towards lower redshift, due to the expansion of the Universe. Numerically, $\Gamma_{\rm dyn}$ is computed by finite difference between halo masses from consecutive snapshots.


In order to prevent H$_2$ cooling, we need $n\Gamma_{\rm dyn}\geq f_{\rm H_2}n_{\rm H}^2\Lambda_{\rm H_2}$. Since, for short enough time-scales (order of Myr at the high redshifts considered in this work), the second term on the right-hand side of Eq.~\eqref{eq:dynheat} is small, it can be neglected, translating the condition into a critical mass accretion rate
\begin{equation}
    \left(\frac{{\rm d} M_{\rm vir}}{{\rm d} t}\right)_{\rm crit}= 0.76\mu f_{\rm H_2} n_{\rm H}\frac{3(\gamma-1)}{2k_{\rm B}}\alpha_{\rm vir}^{-1}\Lambda_{\rm H_2}M_{\rm vir}^{1/3}.
\end{equation}

We follow the evolution of all mini-haloes in the simulation, and track whether at any stage they exceed the minimum mass for H$_2$ cooling.   For any mini-halo that exceeds this threshold, we further check whether they fulfill the dynamical-heating criterion.  For any halo above the H$_2$ cooling minimum mass which, during any step of the evolution, fails the latter criterion, we assume that it remains a plausible host of a direct-collapse BH only for one H$_2$ cooling time $\tau_{\rm H_2}$~\citep{fernandez14}, after which it is assumed to form stars and is excluded from the candidate list, unless the condition for dynamical heating was fulfilled again within the time $\tau_{\rm H_2}$. 
For example, a halo could experience a strong accretion or a merger at a given time, events that result in a dynamical-heating rate much higher that the critical value, then the accretion drops below the threshold for a short time, and then increases again above threshold. If the time during which the halo is below the critical threshold is shorter than the halo H$_2$ cooling time $\tau_{\rm H_2}$, we retain the halo as a candidate.


\subsection{Synchronised pairs}

While dynamical heating may prevent H$_2$ cooling and star formation in the mini-halo stage, it is unclear whether it can help avoid catastrophic cooling (via H atoms and then H$_2$) and the ensuing star formation once a halo crosses the atomic-cooling threshold (ACT), even if the halo is pristine (see more discussion of this below). A plausible scenario proposed to achieve this, and keep even ACHs free of H$_2$ cooling, involves synchronised pairs of ACHs. 
In this scenario, once the first halo crosses the atomic cooling threshold (Halo 1), it cools and forms stars in $\Delta t_{\rm SF} = t_{\rm ff} \equiv \sqrt{(3\upi)/(32{\rm G}\rho)}$, subsequently providing a strong enough LW flux able to prevent the formation of H$_2$ in the companion halo (Halo 2). This allows gas in Halo 2 to collapse nearly isothermally at $T\sim 10^4$~K, leading to the conditions (i.e. large accretion rate) required for the formation of a DCBH. 

In order for this process to occur, in our model we require the time difference $\Delta t$ between the onset of SF in Halo 1, occurring at $t=\tau_{\rm ff,1}$ after the ACT crossing, and the ACT crossing in Halo 2 to be shorter than $\Delta t_{\rm SN}=5$~Myr, the characteristic time after which the first massive stars explode, ejecting metals that enrich the medium~\citep[see, e.g.][]{regan17}. This translates to a maximum time separation between the two ACT crossing events $\tau_{\rm ff,1}<\Delta t < \tau_{\rm ff,1} + \Delta t_{\rm SN}$. In addition, to ensure that the LW flux is large enough to prevent H$_2$ formation, we also require the halo separation to be $200 < \Delta r < 500$~physical pc. 
Since, in our model, we also explore the case of star formation occurring in mini-haloes that had not been sufficiently dynamically heated, in these cases we exclude from the list of synchronised pair candidates all haloes that had formed stars during their mini-halo phase. Nevertheless, below we also compare such results with a simpler approach based on the number of progenitors, as discussed in Section~\ref{sec:pollution}.

\subsection{The LW radiation effect}

An important role in determining the abundance of potential DCBHs is played by the value of $J_{21}$ chosen for the analysis. In this work, we consider three different cases. In the first two, $J_{21}$ is kept constant at 0.01 and 1.0, respectively, which brackets the fiducial values in \citet{trenti09}, and values expected for the mean LW background at the redshifts considered here. In the third, we self-consistently compute the local $J_{21}$ due to nearby star-forming haloes, which can be much larger than the background, depending on the halo's location. We follow \citet{dijkstra14} and compute the local flux impinging on each halo as the sum of the fluxes over all star-forming haloes in the tracked region, with the contribution of each halo given by
\begin{equation}
\label{eq:j21}
    J_{21}=\frac{h\langle\nu\rangle}{\Delta\nu}\frac{f_{\rm esc,LW}Q_{\rm LW}}{16\upi r^2}M_\star,
\end{equation}
where $M_\star=f_\star(\Omega_{\rm b}/\Omega_{\rm m})M_{\rm vir}$.
Here we use $\langle\nu\rangle=2.99\times 10^{15}$~Hz and $\Delta\nu=5.79\times 10^{14}$~Hz as the average frequency and the bandwidth of the LW band,
extending from 11.2 to 13.6~eV,
and $f_{\rm esc,LW}$ is the escape fraction of LW radiation from the halo, which we set to 100 per cent. The latter assumption is reasonable in most cases, in particular far away from the irradiating halo, and when the relative gas velocities are large, hence doppler-shifting the LW photons out of resonance.  On the other hand, the measured LW $f_{\rm esc}$ can drop significantly closer to the halo, if the stellar ionisation front  expands slowly and/or if the stellar spectrum softens, allowing for H$_2$ to form in the outskirts of the