The electron-boson spectral density function of underdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ and YBa$_2$Cu$_3$O$_{6.50}$

We investigate the electron-boson spectral density function, $I^2\chi(\omega,T)$, of CuO$_2$ plane in underdoped Bi$_2$Sr$_2 $CaCu$_2$O$_{8+\delta}$ (Bi-2212) and underdoped YBa$_2$Cu$_3 $O$_{6.50}$ (Y-123) systems using the Eliashberg formalism. We apply a new (in-plane) pseudogap model to extract the electron- boson spectral function. For extracting the spectral function we assume that the spectral density function consists of two components: a sharp mode and the broad Millis-Monien-Pines (MMP) mode. We observe that both the resulting spectral density function and the intensity of the pseudogap show strong temperature dependences: the sharp mode takes most spectral weight of the function and the peak position of the sharp mode shifts to lower frequency and the depth of pseudogap, $1-\tilde {N}(0,T)$, is getting deeper as temperature decreases. We estimate fictitious (maximum) superconducting transition temperatures, $T_c(T)$, from the extracted spectral functions at various measured temperatures using a generalized McMillan formula. The estimated (maximum) $T_c$ also shows a strong temperature dependence; it is higher than the actual $T_c$ at all measured temperatures and decreases with temperature lowering. Since as lowering temperature the pseudogap is getting stronger and the maximum $T_c$ is getting lower we propose that the pseudogap may suppress the superconductivity in cuprates.