Constraints on primordial black holes in the mixed dark matter scenarios using the ratio $\rm (^3{He}+D)/H$

We derive the upper limit on the dark matter (DM) fraction in primordial black holes (PBHs) in the mixed DM scenarios. In this scenarios, a PBH can accrete weakly interacting massive particles (WIMPs) to form a ultracompact minihalo (UCMH) with a density profile of $\rho_{\rm DM}(r)\sim r^{-9/4}$. The energy released from UCMHs due to dark matter annihilation has influence on the photodissociation of $^{4}{\rm He}$, producing the $^{3}{\rm He}$ and the D. By requiring that the ratio $\rm (^3{He}+D)/H$ caused by UCMHs does not exceed the measured value, we derive the upper limit on the dark matter fraction in PBHs. For the canonical value of DM thermally averaged annihilation cross section $\left<\sigma v\right>=3\times 10^{-26}\rm cm^{3}s^{-1}$, we find that the upper limit is $f_{\rm PBH} < 0.35(0.75)$ for DM mass $m_{\chi}=1(10)~\rm GeV$. Compared with other limits obtained by different astronomical measurements, although our limit is not the strongest, we provide a different way of constraining the cosmological abundance of PBHs.