The dilaton as a candidate for dark matter

We examine consequences of the stabilization of the dilaton through the axion. An estimate of the resulting dilaton potential yields a relation between the axion parameter $m_a f_{PQ}$ and the average instanton radius, and predicts the ratio between the dilaton mass $m_ϕ$ and the axion mass $m_a$. If we identify the string axion with a Peccei--Quinn axion, then $m_ϕm_{Pl} \sim m_a f_{PQ}$, and the dilaton should be strongly aligned $\sqrt{}\leq 10^{-4}m_{Pl}$ at the QCD scale, in order not to overclose the universe.