The spectrum of $n_s$ constraints from DESI and CMB data

We present the spectrum of $n_s$ constraints from current CMB data (Planck, ACT, SPT-3G) combined with DESI 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 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 cosmic microwave background (CMB) experiments, the constraints on $n_s$ shift upward when CMB data is combined with the latest baryon acoustic oscillation (BAO) data from the Dark Energy Spectroscopic Instrument (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 Atacama Cosmology Telescope (ACT) DR6 data and DESI DR2 data. Here we present constraints from Planck (PR3 and PR4), ACT, the South Pole Telescope (SPT), and the combination of all three CMB experiments, 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 $\tau$.