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

TL;DR

The paper analyzes the spectrum of constraints on the scalar spectral index $n_s$ using current CMB data from Planck, ACT, SPT-3G, and their combination CMB-SPA, with and without DESI DR2 BAO data, and investigates the interplay with the reionization optical depth $\tau$. It demonstrates a consistent upward shift in $n_s$ when DESI BAO is included across datasets, with the largest shift for ACT, and shows strong correlations between $n_s$ and BAO parameters $r_d h$ and $\Omega_m$, influenced by the $n_s$–$\omega_m$ degeneracy and lensing effects. When DESI is added, the $n_s$ constraints, especially for the CMB-SPA combination, disfavour several inflation models (e.g., Higgs, Starobinsky, exponential $\alpha$-attractors) at more than $2\sigma$, indicating the BAO-CMB tension as a driving factor behind the shift. The work underscores the importance of understanding BAO-CMB tension and the $n_s$–$\tau$ interplay, and advocates for future data and analyses (e.g., Euclid, DESI reanalyses) to refine $n_s$ constraints and inflation-model discrimination.

Abstract

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 $τ$.

Figures (4)

• Figure 1: CMB constraints on $n_s$ and the BAO parameters $r_d h$ and $\Omega_m$. We consider CMB data from Planck PR4, ACT, SPT, and the combination CMB-SPA, with and without DESI BAO data. (See Sec. \ref{['sec:datasets']} for the details of each dataset). In each case, the inclusion of DESI leads to a shift in $n_s$ due to the correlation of $n_s$ with the BAO parameters in the fit to CMB data.
• Figure 2: Interplay with $\tau$: CMB constraints on $\tau$, the spectral index $n_s$, and the BAO parameters $r_d h$ and $\Omega_m$. We consider CMB data from Planck, ACT, SPT, and the combination CMB-SPA, with and without DESI BAO data. Paralling $n_s$, in each case, the inclusion of DESI leads to a shift in $\tau$ due to the correlation of $\tau$ with the BAO parameters in the fit to CMB data.
• Figure 3: Influence of an artificial $\tau$ prior on ACT. The parameter shifts mirror that from the addition of DESI, and the $n_s$ constraint matches identically.
• Figure 4: Constraints with and without CMB lensing. We consider P-ACT, SPT, SPT+DESI, and CMB-SPA+DESI. The inclusion of lensing makes a negligible change to the $n_s$ constraint. In the case of SPT, lensing lessens the correlation between $n_s$ and BAO parameters, but removes most of the overlap with DESI and SPT contours, indicating an increase in the BAO tension. This leads to a slight relative increase in $n_s$ when SPT is combined with DESI with CMB lensing included.