Inflation at the End of 2025: Constraints on $r$ and $n_s$ Using the Latest CMB and BAO Data

Abstract

Inflation elegantly provides initial conditions for the standard model of cosmology, while solving the horizon, flatness, and magnetic monopole problems. Inflationary models make predictions for the tensor-to-scalar ratio $r$ and the spectral index $n_s$ of initial density fluctuations. In light of relevant data releases this year, we present constraints on these two parameters using the latest cosmic microwave background (CMB) and baryon acoustic oscillation data (BAO) available. Using data from Planck, the South Pole Telescope, Atacama Cosmology Telescope, and BICEP/Keck experiments, we derive $n_s=0.9682\,\pm\,0.0032$ and a 95% upper limit of $r<0.034$. This upper limit on $r$ is consistent with the official BICEP/Keck result given the numerical precision of the analyses and our choice to impose the self-consistency relation for single field slow-roll inflation on the tensor power spectrum; the $r$ constraint is not impacted by the additional CMB data. While adding DESI BAO data to the CMB data has a negligible impact on $r$, the $n_s$ constraint shifts upward to $0.9728\,\pm\,0.0029$, which favours monomial inflaton potentials with $N_\star\sim 50$ over Starobinsky $R^2$ or Higgs inflation with $N_\star = 51$ and $N_\star = 55$, respectively. This shift is caused by marginally significant differences between the CMB and DESI data that remain unexplained in the context of the standard model. We show that a class of polynomial $α$-attractor models can predict the CMB and CMB+DESI $n_s$ results with $N_\star=47.1$ and $N_\star=55.1$, respectively. While future data will improve our sensitivity to $r$, robust $n_s$ constraints are just as crucial to differentiate between inflation models. We make the data needed to reproduce the new CMB and BAO results and visualisation tools for $r$-$n_s$ figures to compare to any inflation model available https://github.com/Lbalkenhol/r_ns_2025 .

Figures (2)

• Figure 1: Constraints on the tensor-to-scalar ratio $r$ and the scalar spectral index $n_\mathrm{s}$ from different cosmological data sets alongside theoretical predictions. The orange contours are derived from Planck and BICEP/Keck data, whereas the blue contours additionally use SPT and ACT data. Filled contours use only CMB data, whereas line contours add BAO data (SDSS for orange contours, DESI for blue contours). We show the prediction for monomial inflation models with different exponents in red; the grey shaded region corresponds to a duration of inflation of $N_{\star}$$=47$-$57$ e-folds for monomial potentials. The predictions of Higgs inflation for $N_{\star}{}=55$ and Starobinsky $R^2$ inflation for $N_{\star}{}=51$ are shown as a white circle and a black square, respectively. The grey dotted lines correspond to the predictions of a polynomial $\alpha$-attractor model with potential $V(\varphi) = V_0 |\varphi|^2/(\mu^2 + |\varphi|^2)$ for $N_{\star}=47$-$57$ (left to right dotted line). We divide the $r$-$n_\mathrm{s}$ plane into regions of concave and convex inflaton potentials. The code to reproduce this figure with SPA+BK(+DESI) contours and all shown theoretical predictions is provided at https://github.com/Lbalkenhol/r_ns_2025.
• Figure 2: Constraints from SPA+BK on $r$ and $n_\mathrm{s}$ (blue contours, same as in Figure \ref{['fig:rns_data']}). Alongside the data constraints we show two forecasts for CMB constraints that may be achieved in the next decade ($10^3 r{}=3\,\pm\,1,\,\sigma(n_\mathrm{s})=2\times 10^{-3}$) at different values of $n_\mathrm{s}$: one at the central value of SPA+BK (black solid contours) and one at the central value of SPA+BK+DESI (black dashed contours). The theoretical predictions shown are the same as in Figure \ref{['fig:rns_data']}.