Current and Future Constraints on the Primordial Power Spectrum

Abstract

The primordial scalar power spectrum provides a powerful window onto early-universe physics, including a broad class of inflationary scenarios. These models often predict a characteristic scalar spectral index $(n_s)$, nonzero running $({\alpha }_s)$, or more general deviations from a pure power-law form. We find that freeing the number of light relic species $(N_{\mathrm{eff}})$ and the sum of the neutrino masses $(\sum m_{\nu })$ broadens the $n_s$ error contours enough that key early-Universe models are no longer excluded. In particular, while both Starobinsky inflation and the Bi-thermal Big Bang model are ruled out at about the 95% confidence level in the six-parameter $\Lambda$CDM model, we find that in the nine-parameter $\Lambda \mathrm{CDM}+{\alpha }_s+N_{\mathrm{eff}}+\sum m_{\nu }$ model, CMB data from Planck, ACT, and SPT (CMB-PAS) combined with DESI DR2 give $n_s=0.9758\pm 0.0064$ and ${\alpha }_s=0.0080\pm 0.0064$, consistent with both models. Future CMB facilities will sharply improve constraints on $n_s$, ${\alpha }_s$, and, more generally, the binned primordial power spectrum $\mathcal{P}(k)$. For example, CMB-PAS constrains $e^{-2\tau }\mathcal{P}(k=0.2{\mathrm{Mpc}}^{-1})$ to 0.5%, while SO-like and CMB-HD-like surveys would tighten this to 0.1%; CMB-HD would also extend this measurement to smaller, previously unprobed scales, constraining $e^{-2\tau }\mathcal{P}(k=30{\mathrm{Mpc}}^{-1})$ to within a factor of ten. We release our forecast code and update the public CMB-HD likelihood and Fisher codes to support CLASS in addition to CAMB.

28 pages, 11 figures, 7 tables. The code used here to generate the current and projected constraints is available at https://github.com/CMB-HD/hdInitPk , and the CMB-HD likelihood and Fisher codes at https://github.com/CMB-HD/hdfisher and https://github.com/CMB-HD/hdlike are updated to support CLASS