Dependence of density perturbations on the coupling constant in a simple model of inflation

In the standard inflationary scenario with inflaton potential $V\left(\Phi \right)=M^4-\frac{1}{4}\lambda {\Phi }^4$, the resulting density perturbations $\frac{\delta \rho }{\rho }$ are proportional to ${\lambda }^{\frac{1}{2}}$. Upper bounds on $\frac{\delta \rho }{\rho }$ require $\lambda \lesssim {10}^{-13\mathrm{}}$. Ratra has shown that an alternative treatment of reheating results in $\frac{\delta \rho }{\rho \propto {\lambda }^{-1\mathrm{}}}$, so that an upper bound on $\frac{\delta \rho }{\rho }$ does not put an obvious upper bound on $\lambda$. We verify that $\frac{\delta \rho }{\rho \propto {\lambda }^{-1\mathrm{}}}$ is indeed a possibility, but show that $\lambda \lesssim {10}^{-13\mathrm{}}$ is still required.