Reply to "Comment on: Cosmological black holes are not described by the Thakurta metric"

In this reply, we address the comment [arXiv:2105.14908] to our recent paper [arXiv:2105.09328], where we argued that the Thakurta metric does not describe cosmological black holes. We clarify that the mass growth of Thakurta black holes is due to an influx of energy (i.e. accretion), which, by definition, is not a feature of geometry. The conclusions of [arXiv:2105.09328] are independent of the interpretation of this energy flux. We show that the average energy density of primordial Thakurta black holes scales as $a^{-2}$ and requires an unrealistic and fine-tuned energy transfer from a smooth dark matter component to the primordial black hole sector.