We present the spectrum of $n_s$ constraints from current cosmic microwave background (CMB) data [Planck, Atacama Cosmology Telescope (ACT), and South Pole Telescope (SPT)-3G] combined with Dark Energy Spectroscopic Instrument (DESI) baryon acoustic oscillation (BAO) data, and highlight the interplay of $n_s$ with the optical depth to reionization $\tau$. The spectral index $n_s$ of the primordial power spectrum provides a window into the early Universe, and constraints on $n_s$ play an important role in discriminating early Universe models such as models of cosmic inflation. Historically constrained by CMB experiments, the constraints on $n_s$ shift upward when CMB data are combined with the latest BAO data from the DESI. Recent work explained the origin of this and the relation to the BAO-CMB tension between CMB experiments and DESI BAO, and as a case study presented constraints on $n_s$ from the combination of ACT DR6 data and DESI DR2 data. Here we present constraints from Planck (PR3 and PR4), ACT, the SPT, and the combination of all three CMB experiments, CMB-SPT + Planck + ACT (CMB-SPA), with and without DESI DR2 BAO data, and with and without CMB lensing data. In all cases the constraint on $n_s$ is shifted upwards when DESI is included, with the largest shift exhibited by ACT. This is accompanied by a commensurate shift in the constraint on the optical depth to reionization $\tau$, which is again greatest for ACT. When CMB data are combined into CMB-SPA and combined with DESI the $n_s$ constraint disfavors at more than $2\sigma$ the inflation models preferred by Planck alone, such as Higgs, Starobinsky, and exponential $\alpha$-attractors, in favor of other models, such as polynomial $\alpha$-attractors. This work motivates the further study of the tension between CMB and DESI BAO data, and of the rich interplay between $n_s$ and $\tau$.

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• Cosmological parameters