Cosmological Constraints from the eBOSS Lyman-$\alpha$ Forest using the PRIYA Simulations

We present new cosmological parameter constraints from the eBOSS Lyman-$\alpha$ forest survey. We use a new theoretical model and likelihood based on the PRIYA simulation suite. PRIYA is the first suite to resolve the Lyman-$\alpha$ forest in a ($120$~Mpc/h~)${}^3$ volume, using a multi-fidelity emulation technique. We use PRIYA to predict Lyman-$\alpha$ forest observables with $\lesssim 1\mathrm{\%}$ interpolation error over an $11$ dimensional ($9$ simulated, $2$ in post-processing) parameter space. We identify an internal tension within the flux power spectrum data. Once the discrepant data is removed, we find the primeval scalar spectral index measured at a pivot scale of $k_0=0.78$ Mpc${}^{-1}$ to be $n_P=1.009_{-0.018}^{+0.027}$ at 68% confidence. This measurement from the Lyman-$\alpha$ forest flux power spectrum alone is in reasonable agreement with Planck, and in tension with earlier eBOSS analyses. The amplitude of matter fluctuations is ${\sigma }_8=0.733_{-0.029}^{+0.026}$ at 68% confidence, in agreement with Dark Energy Survey weak lensing measurements and other small-scale structure probes and in tension with CMB measurements from Planck and ACT. The effective optical depth to Lyman-$\alpha$ photons from our pipeline is in good agreement with earlier high resolution measurements. We find a linear power at $z=3$ and $k=0.009$ s/km of ${\mathrm{\Delta }}_L^2=0.302_{-0.027}^{+0.024}$ with a slope $n_{\mathrm{e}\mathrm{f}\mathrm{f}}=-2.264_{-0.018}^{+0.026}$. Our flux power spectrum only chains prefer a low level of heating during helium reionization. When we add IGM temperature data we find $n_P=0.983\pm 0.020$ and ${\sigma }_8=0.703_{-0.027}^{+0.023}$. Our chains prefer an early and long helium reionization event, as suggested by measurements from the helium Lyman-$\alpha$ forest. In the near future we will use our pipeline to infer cosmological parameters from the DESI Lyman-$\alpha$ data.