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

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 .

💡 Research Summary

This paper presents a comprehensive analysis of the most recent cosmological observations available at the end of 2025, focusing on the two key inflationary parameters: the tensor‑to‑scalar ratio r and the scalar spectral index nₛ. The authors combine data from the Planck satellite (both PR3 and the newer PR4 releases), the South Pole Telescope (SPT‑3G D1), the Atacama Cosmology Telescope (ACT DR6), and the BICEP/Keck B‑mode measurements, forming a joint “SPA+BK” likelihood that includes temperature, E‑mode polarization, lensing, and B‑mode information. In addition, they incorporate the second data release of the Dark Energy Spectroscopic Instrument (DESI DR2) BAO measurements, which provide precise distance information at low redshift.

The theoretical framework is the standard ΛCDM model extended by a single additional parameter r, with the tensor spectral tilt nₜ fixed by the single‑field slow‑roll consistency relation nₜ = −r/8. The scalar amplitude log(10¹⁰Aₛ) and the spectral index nₛ are sampled together with the usual six ΛCDM parameters (Ω_bh², Ω_ch², H₀, τ_reio). A Gaussian prior on the reionisation optical depth (τ ≈ 0.051 ± 0.0062) is imposed, reflecting Planck low‑ℓ polarization constraints. Uniform priors are used for all other parameters. Parameter inference is performed with the Cobaya MCMC sampler, employing the CLASS Boltzmann code for theoretical spectra. Convergence is required at Gelman‑Rubin R − 1 < 0.02, and results are visualised with GetDist.

Results without BAO (SPA+BK):

• Spectral index: nₛ = 0.9682 ± 0.0032 (68 % confidence).
• Tensor‑to‑scalar ratio: r < 0.034 (95 % upper limit).
  The r limit is driven almost entirely by the BICEP/Keck B‑mode data; adding SPT and ACT data does not noticeably tighten the bound. The nₛ value reflects the combined power of Planck, ACT, and SPT, each contributing information on different angular scales. The authors note that the self‑consistency relation for nₜ slightly raises the expected B‑mode power compared with analyses that fix nₜ = 0, thereby marginally lowering the r upper bound.

Results with DESI BAO (SPA+BK+DESI):

• Spectral index: nₛ = 0.9728 ± 0.0029 (68 % confidence).
• Tensor‑to‑scalar ratio: r < 0.035 (95 % upper limit).
  The inclusion of DESI BAO data leaves the r constraint essentially unchanged but shifts the central nₛ value upward by about 0.0046, a statistically significant movement given the quoted uncertainties. This shift mirrors similar findings in recent literature and suggests a mild tension between high‑redshift CMB measurements and low‑redshift BAO distances. The authors explore the dependence on the τ prior: relaxing the τ prior yields τ = 0.076 ± 0.014 and pushes nₛ to 0.9722 ± 0.0041, confirming the known correlation between τ and nₛ.

Theoretical interpretation:
The paper compares the observational constraints with four representative families of inflationary models: (i) monomial potentials V ∝ φⁿ with n = 1, 2/3, 1/3, evaluated for 47 ≤ N⋆ ≤ 57 e‑folds; (ii) Starobinsky R² inflation (N⋆ = 51); (iii) Higgs‑inflation (N⋆ = 55); and (iv) polynomial α‑attractor models with V = V₀|φ|²/(μ² + |φ|²) (k = 2) across the same e‑fold range. Using the standard slow‑roll expressions, the authors find that the CMB‑only nₛ = 0.9682 is compatible at the 2 σ level with a monomial model of N⋆ ≈ 47 (corresponding to an effective power n ≈ 0.33) and is within 2 σ of the Starobinsky prediction, while Higgs inflation lies at 1.3 σ. When DESI is added, the higher nₛ = 0.9728 aligns almost perfectly with the α‑attractor prediction for N⋆ ≈ 55.1, and remains within 1.6 σ of the N⋆ = 47 monomial case. Consequently, the current data modestly favour models with a slightly redder (larger) spectral index, i.e., those with a more concave potential shape, over the classic Starobinsky or Higgs scenarios.

Future prospects:
Figure 2 in the manuscript projects the sensitivity of next‑generation CMB experiments (CMB‑S4, LiteBIRD, etc.) that could achieve σ(r) ≈ 3 × 10⁻³ and σ(nₛ) ≈ 2 × 10⁻³. The authors illustrate two forecast contours centred on the present CMB‑only and CMB + DESI best‑fit nₛ values, showing that forthcoming data will be able to discriminate between the monomial, Starobinsky, Higgs, and α‑attractor families at the few‑σ level.

Data and code release:
All MCMC chains, likelihood configurations, and plotting scripts are publicly available on GitHub (https://github.com/Lbalkenhol/r_ns_2025) together with an interactive Streamlit application (https://r-ns-plot.streamlit.app/). This openness enables other researchers to overlay arbitrary inflationary predictions on the r‑nₛ plane and to update the constraints as new data become available.

Conclusion:
The analysis confirms that, as of late 2025, the tensor‑to‑scalar ratio remains tightly bounded at r < 0.034, with no significant improvement from the addition of DESI BAO data. However, the scalar spectral index is now measured with sub‑percent precision, and its modest upward shift when BAO data are included has non‑trivial implications for model selection. Polynomial α‑attractor models with N⋆ ≈ 55 provide the best overall fit, while classic Starobinsky and Higgs inflation are slightly disfavoured but still within the 2 σ envelope. The authors emphasize that future high‑precision CMB polarization measurements will be essential to push r limits down to the 10⁻³ level, while continued improvements in nₛ will further sharpen the discrimination among competing inflationary scenarios.