Warm Dark Matter meets Cold Dark Matter Isocurvature

Isocurvature fluctuations can be generated in various scenarios in the early Universe. In particular, some specific models predict those with a blue-tilted spectrum, which is consistent with the constraints from cosmic microwave background such as Planck, although isocurvature fluctuations with an almost scale-invariant spectrum are severely constrained. We argue that cold dark matter (CDM) isocurvature fluctuations with blue-tilted spectrum are not only consistent with current cosmological data, but also can loosen the bound on the masses of warm dark matter (WDM), which suppresses small-scale power. In pure thermal WDM models with the adiabatic initial condition, a combination of the data from Lyman-$\alpha$, gravitational lensing, and Milky Way satellites gives a lower bound on the WDM mass as $6~{\rm keV}$ at $95\%$ C.L. while mixed WDM+CDM models loosen these bounds to $m_{\rm WDM}\sim1$ keV for a warm-fraction $f_{\rm WDM}\lesssim0.14$ and $m_{\rm WDM}\sim600$ eV for $f_{\rm WDM}\lesssim0.08$. On the other hand, as we demonstrate, WDM scenarios with a blue-tilted CDM isocurvature power spectrum, even with only $1\%$ CDM contribution ($f_{\rm WDM}\sim0.99$), can allow WDM masses as low as $600$ eV. We further assess the implications of this ``warm + cold-isocurvature'' extension for the small-scale structure by performing $N$-body simulations, particularly focusing on nonlinear matter power spectrum and halo mass function.