tokens: either N prog 2 or N prog 5, which correspond to at most 10 or 40 per cent probability of being star forming during the mini-halo phase, respectively. We obtained results from this simple approach as an academic exercise, and in order to be able to compare results with previous work by Dijkstra et al. ( 2014 ). In the second approach, instead, which we employ as our fiducial case, we self-consistently account for SF in mini-haloes that can cool via H 2 , and exclude these from the list of candidate DH mini-haloes. For environmental pollution, we follow all haloes able to form stars in our cosmological volume, i.e. mini-haloes not DH and all ACHs, and compute the size of the metal bubble starting from the time of the first SN explosion. Hence, the metal bubbles start to expand in our model Δ t = t ff + Δ t SN after the halo is first declared star-forming. To model SF, we assume that the stellar mass formed in ACHs is f ⋆ = 0.05 of the baryonic mass, translating to M ⋆ = f ⋆ Ω b /Ω m M vir . For mini-haloes, instead, we consider a lower star-forming efficiency f ⋆ = 0.005. In both cases, we assume that SF starts with a burst as soon as one free-fall time (defined at the maximum density achievable via adiabatic contraction ρ gas = η gas ρ max ) has elapsed since (i) dynamical heating conditions have failed or (ii) the ACT has been exceeded (if no previous SF in the mini-halo had already occurred), and then proceeds smoothly following the increase in stellar mass with time in the halo. Following Weaver et al. ( 1977 ) and Madau, Ferrara Rees ( 2001 ), we define the bubble radius as where is the average SN luminosity associated to the stellar population in the halo, t the time since the burst of SF, and n ≡ Δ gas Ω b ρ crit (1 + z ) 3 / m H is the gas density (in cm −3 ) the bubble expands through. Now, we assume for simplicity that L SN ≈ E SN η SN M ⋆ ( t )/ t , where is the energy released per SN, M ⋆ ( t ) = f ⋆ (Ω b /Ω m ) M vir ( t ) is the stellar mass at time t , and is the number of SNe per stellar mass formed, and replace them in the bubble radius equation, we obtain, for t > Δ t SN , We notice that this equation reduces, for t ≫ Δ t SN , to the equation reported in Dijkstra et al. ( 2014 ), apart from the slightly lower normalization. In Dijkstra et al. ( 2014 ), the baryon overdensity through which the bubble expands is assumed to be Δ gas ≈ 60, typical of the overdensities found within haloes (60 ≲ Δ gas ≲ 1000), although the bubble spends most of the time in the intergalactic medium at Δ gas ∼ 1−10. This results in the bubble size being underestimated (Dijkstra et al. 2014 ; Habouzit, Volonteri Dubois 2017 ). In our model, instead, we also account how the overdensity changes outside the halo, by fitting the asymptotic case of an extremely rare halo with from Barkana ( 2004 ), and simply express Δ gas as where x ≡ log 10 ( R / R vir ), R is the bubble radius, and the factor 0.787 has been chosen to join the inner constant value and
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