  A period of early matter domination can give rise to the correct dark matter
abundance for a broad range of dark matter annihilation rate $\langle
\sigma_{\rm ann} v \rangle_{\rm f}$. Here, we examine this scenario for
situations where $\langle \sigma_{\rm ann} v \rangle_{\rm f}$ is below the
nominal value for thermal dark matter $3 \times 10^{-26}$ cm$^3$ s$^{-1}$ as
possibly indicated by some recent experiments. We show that obtaining the
correct relic abundance sets a lower bound on the duration of early matter
domination era in this case. On the other hand, provided that the
post-inflationary universe has an equation of state characterized by $w \leq
1/3$, the requirement that the scalar spectral index $n_s$ be within the
observationally allowed range limits the duration of this epoch from above. By
combining these considerations, we show that the current and future cosmic
microwave background experiments can tightly constrain the parameter space for
this scenario. In particular, models of inflation with a tensor-to-scalar ratio
below ${\cal O}(0.01)$ may disfavor non-thermal supersymmetric dark matter from
a modulus-driven early matter domination epoch.
