Lyman-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math> constraints on ultralight scalar dark matter: Implications for the early and late universe

We investigate constraints on scalar dark matter (DM) by analyzing the Lyman-$\ensuremath{\alpha}$ forest, which probes structure formation at medium and small scales, and also by studying its cosmological consequences at high and low redshift. For scalar DM that constitutes more than 30% of the total DM density, we obtain a lower limit $m\ensuremath{\gtrsim}{10}^{\ensuremath{-}21}\text{ }\text{ }\mathrm{eV}$ for the mass of scalar DM. This implies an upper limit on the initial field displacement (or the decay constant for an axion-like field) of $\ensuremath{\phi}\ensuremath{\lesssim}{10}^{16}\text{ }\text{ }\mathrm{GeV}$. We also derive limits on the energy scale of cosmic inflation and establish an upper bound on the tensor-to-scalar ratio of $r&lt;{10}^{\ensuremath{-}3}$ in the presence of scalar DM. Furthermore, we show that there is very little room for ultralight scalar DM to solve the ``small-scale crisis'' of cold DM without spoiling the Lyman-$\ensuremath{\alpha}$ forest results. The constraints presented in this paper can be used for testing generic theories that contain light scalar fields.