This study sets new constraints on $\mathrm{cold}+\mathrm{warm}$ dark matter (CWDM) models by leveraging the small-scale suppression of structure formation imprinted in the Lyman-$\alpha$ forest. Using the Sherwood-Relics suite, we extract high-fidelity flux power spectra from simulated Lyman-$\alpha$ forest data, spanning a broad range of cosmologies and thermal histories. This enables precise constraints on the warm dark matter (WDM) fraction, $f_{\mathrm{WDM}}$, and the mass of the WDM particle, $m_{\mathrm{WDM}}$. A key advancement of our analysis is the integration of a neural network emulator directly at the likelihood level, significantly accelerating Bayesian parameter inference. With new observations of high-redshift ($z=4.2–5.0$) quasar spectra from UVES and HIRES, we establish stringent upper limits: for $m_{\mathrm{WDM}}=1\mathrm{keV}$, we find $f_{\mathrm{WDM}}<0.16$ ($2\sigma$), with constraints loosening to 35%, 50%, and 67% for $m_{\mathrm{WDM}}=2,3$, and 4 keV, respectively. Our results for pure WDM reaffirm the lower bounds of previous work. Crucially, we account for the fixed resolution of simulations and the impact of patchy reionization, demonstrating their minimal influence on mixed dark matter constraints. This robustness paves the way for tighter bounds with improved statistical samples in the future. Our findings suggest that CWDM models can naturally accommodate mild suppression of matter clustering in the high-redshift Lyman-$\alpha$ forest 1D flux power, potentially offering a resolution to some of the ongoing cosmological tensions at low redshifts, namely the $S_8$ tension.

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• Cosmological parameters
• Cosmology
• Dark matter
• Large scale structure of the Universe
• Particle dark matter
• Astrophysical & cosmological simulations