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 $τ$. 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 $τ$, 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 $2σ$ the inflation models preferred by Planck alone, such as Higgs, Starobinsky, and exponential $α$-attractors, in favor of other models, such as polynomial $α$-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 $τ$.

💡 Research Summary

This paper presents a comprehensive analysis of how the inclusion of the latest Dark Energy Spectroscopic Instrument (DESI) baryon acoustic oscillation (BAO) measurements affects constraints on the scalar spectral index (n_s) and the optical depth to reionization (τ) obtained from current cosmic microwave background (CMB) experiments. The authors consider four CMB data sets—Planck (PR4), Atacama Cosmology Telescope (ACT) DR6, South Pole Telescope (SPT‑3G) DR1, and a combined “CMB‑SPA” data set that merges all three—and examine each both with and without DESI DR2 BAO data. Parameter inference is performed with Cobaya, using Markov‑Chain Monte Carlo sampling and a stringent convergence criterion ((R-1<0.01)).

Key findings are: (1) In all CMB data sets, (n_s) is positively correlated with the BAO parameters (r_d h) and (\Omega_m). The strength of this correlation depends on the experiment’s sensitivity to large angular scales; ACT and Planck, which have substantial low‑ℓ information, show the strongest correlation, while SPT, focused on smaller scales, shows the weakest. (2) Adding DESI BAO data systematically shifts the posterior of (n_s) upward. The shift is modest for Planck ((n_s=0.9690\pm0.0035) versus (0.9638\pm0.0040) without DESI) but pronounced for ACT ((n_s=0.9767\pm0.0068) versus (0.9666\pm0.0076)). The combined CMB‑SPA + DESI yields (n_s=0.9737\pm0.0025), closely tracking the Planck‑plus‑ACT trend. (3) The optical depth (τ) moves in tandem with (n_s) because an increase in (τ) suppresses high‑ℓ power, which can be compensated by a larger (n_s). Consequently, ACT experiences the largest (τ) shift ((0.0562) → (0.0636)), while Planck’s shift is smaller ((0.0516) → (0.0554)). An artificial Gaussian prior on (τ) ((\mu=0.11,\sigma=0.011)) reproduces the ACT + DESI (n_s) shift, confirming the causal link.

The authors then confront inflationary model predictions. Models such as Higgs inflation, Starobinsky inflation, and exponential (\alpha)-attractors predict (n_s = 1 - 2/N_) with (N_) in the range 50–60, corresponding to (n_s) between 0.9600 and 0.9667. While Planck‑only data are fully compatible with this range, the CMB‑SPA + DESI combination pushes the posterior well above it, disfavoring these benchmark models at >2σ and favoring polynomial (\alpha)-attractor scenarios instead.

The paper emphasizes that the observed “BAO‑CMB tension”—a 2–3σ discrepancy between DESI BAO measurements and each CMB experiment—drives the correlated shifts in (n_s) and (τ). The tension is strongest for ACT (≈3.1σ) and weakest for Planck (≈2.0σ). The authors argue that resolving this tension, either through improved BAO measurements or refined low‑ℓ CMB polarization data, is essential for robustly testing inflationary physics.

In conclusion, the study quantifies how BAO data reshape early‑universe parameter constraints, highlights the intertwined nature of (n_s) and (τ), and underscores the need for coordinated CMB‑BAO analyses in the era of next‑generation experiments such as CMB‑S4 and LiteBIRD.