Predictions on the abundance of primordial black holes: results from the SMILE VLBI milli-lensing sample

Abstract: Primordial black holes (PBHs) formed in the early universe represent a well-motivated dark matter candidate. However, their abundance in the intermediate-to-high mass range ($10^6$-10<sup>9 M⊙10<sup>9\,M_\odot) remains relatively less constrained. In this mass regime, the corresponding Einstein radii on the milliarcsecond scale make Very Long Baseline Interferometry (VLBI) the most direct observational probe via milli-lensing of compact radio sources. In this paper, we present predictions for the abundance of PBH dark matter ($f_{\rm PBH}$) using the large SMILE sample of $\sim 5000$ flat-spectrum compact radio sources. Under the optimistic observational conditions (with an angular resolution of $1.5\,\mathrm{mas}$, a maximum image separation of $100\,\mathrm{mas}$, and a maximum detectable flux ratio of $40$), assuming that no milli-lensing events are confirmed, we obtain the projected 95% upper limit of $f_{\rm PBH}\lesssim0.12$ in the intermediate-to-high mass range, which would be approximately an order of magnitude tighter than previous VLBI milli-lensing limits. Broader lognormal mass functions yield correspondingly weaker projected limits. Generalising the optical depth to account for magnification bias and finite-core resolvability changes these projected limits by only about $5$%. Moreover, we show that the predicted limits are highly sensitive to a small number of confirmed events, i.e., even one confirmed milli-lensing event would relax the projected upper limit to $0.20$%, highlighting the critical importance of robust candidate confirmation in the SMILE sample. Finally, under the optimistic baryonic accretion scenario, the PBH mass growth changes the projected limits by $\lesssim 1$% within the VLBI-sensitive mass range, confirming the robustness of our results.