Large isotope effect on $T_c$ in cuprates despite of a small electron-phonon coupling

We calculate the isotope coefficients $\alpha$ and $\alpha^\ast$ for the superconducting critical temperature $T_c$ and the pseudogap temperature $T^\ast$ in a mean-field treatment of the t-J model including phonons. The pseudogap phase is identified with the $d$-charge-density wave ($d$-CDW) phase in this model. Using the small electron-phonon coupling constant $\lambda_d \sim 0.02$ obtained previously in LDA calculations in YBa$_2$Cu$_3$O$_7$, $\alpha^{\ast}$ is negative but negligible small whereas $\alpha$ increases from about 0.03 at optimal doping to values around 1 at small dopings in agreement with the general trend observed in many cuprates. Using a simple phase fluctuation model where the $d$-CDW has only short-range correlations it is shown that the large increase of $\alpha$ at low dopings is rather universal and does not depend on the existence of sharp peaks in the density of states in the pseudogap state or on specific values of the phonon cutoff. It rather is caused by the large depletion of spectral weight at low frequencies by the $d$-CDW and thus should also occur in other realizations of the pseudogap.