Dependence of Density Perturbations on the Coupling Constant in a Simple Model of Inflation

In the standard inflationary scenario with inflaton potential $V(\Phi)=M^4-{1\over4}\lambda\Phi^4$, the resulting density perturbations $\delta\rho/\rho$ are proportional to $\lambda^{1/2}$. Upper bounds on $\delta\rho/\rho$ require $\lambda < 10^{-13}$. Ratra has shown that an alternative treatment of reheating results in $\delta\rho/\rho \propto \lambda^{-1}$, so that an upper bound on $\delta\rho/\rho$ does not put an obvious upper bound on $\lambda$. We verify that $\delta\rho/\rho \propto \lambda^{-1}$ is indeed a possibility, but show that $\lambda < 10^{-13}$ is still required.