Constraints on primordial black holes with CMB spectral distortions

In the mixed dark matter scenarios consisting of primordial black holes (PBHs) and weakly interacting massive particles (WIMPs), WIMPs can be accreted onto PBHs to form ultracompact minihalos (UCMHs) with a density spike in the early universe. Compared with the classical dark matter halo, UCMHs are formed earlier and have a higher density of center. Since the annihilation rate is proportional to the squared number density of WIMPs, it is expected that WIMPs annihilation within UCMHs is enhanced and has influences on the early universe. Between the time of recombination and matter-radiation equality, the energy released from WIMPs annihilation within UCMHs is injected into the Universe resulting in CMB $y$-type distortion. We investigate these effects and derive the upper limits on the abundance of PBHs taking advantage of the observational results of Far Infrared Absolute Spectrophotometer (FIRAS). We find that for the WIMPs mass range $1\le m_{\chi}\le 1000~\rm GeV$, the upper limits on the abundance of PBHs are $5\times 10^{-3}\le \Omega_{\rm PBH}\le 5\times 10^{-2}$.