(\[ours\]) is by plotting 2(V\^)\^[2]{}- V\^V\^-  ()\^2 Then eq.(\[ours\]) means $\chi\geq 0$. We plot in fig.(\[starobinsky\])-a that quantity for the above potential with $A=1$ and several values of $B$. We see that the bound seems to be obeyed. In fact one can check that above $b\simeq 2$ the bound is slightly violated at the per-mil label, something not visible in the figure. However we do not have control of the theory to that level and one may say that this model passes the test. There are several other schemes leading to linear potentials which we will not discuss here, see e.g. [@McAllister:2008hb; @Higgsotic].

More generally one may consider monomial potentials of the form $V=\phi^a$, $a\geq 0$. For them the condition $\chi \geq 0$ gives $(a-1)(a-2) M^2_\text{p} -\phi^2 \geq 0$. For $0\leq a <1$ the potential has only tiny violations of the bound at small $\phi$, in the region $\phi < \sqrt{(a-1)(a-2)} M_\text{p} $. The same formula applies for $a>2$, here the violations are large but are trans-Planckian for $a>2.7$. However, for $1<a\leq2$ the bound is irremediably violated at all points of field space. Finally, $a=0$ and $a=1$ are the only pure monomials which satisfy the bound at all points of 