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Current and Future Constraints on the Primordial Power Spectrum

This paper demonstrates that extending the standard cosmological model to include running spectral index, extra relativistic species, and neutrino masses relaxes constraints on the primordial power spectrum, thereby reviving previously excluded inflationary models, while forecasting that future CMB experiments will significantly tighten measurements of spectral parameters and probe smaller scales than currently possible.

Original authors: Zachary Cheslog, Emily Finson, Amanda MacInnis, Neelima Sehgal, Niayesh Afshordi, Simran K. Nerval, Renée Hložek

Published 2026-09-30

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CC BY 4.0

Original authors: Zachary Cheslog, Emily Finson, Amanda MacInnis, Neelima Sehgal, Niayesh Afshordi, Simran K. Nerval, Renée Hložek

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To understand the universe as it is today, cosmologists must look back to its very first moments. Roughly 13.8 billion years ago, the cosmos underwent a period of rapid expansion known as inflation, a fraction of a second during which the universe grew from something smaller than an atom to something vast and smooth. This explosive beginning left behind a faint afterglow, a bath of light called the cosmic microwave background, which fills the sky in every direction. Within this ancient light lies a map of the universe's earliest seeds: tiny fluctuations in density that would eventually grow into the galaxies and stars we see today. The shape and size of these fluctuations are described by something called the primordial power spectrum. By measuring this spectrum with extreme precision, scientists can test different theories about how inflation happened, much like a geologist reading rock layers to understand the history of the Earth. If the measurements match a specific prediction, a theory gains strength; if they do not, the theory may need to be discarded.

A team of researchers has recently combined the most powerful current observations of this ancient light with the latest data on the distribution of galaxies to refine our understanding of these early moments. They analyzed data from three major telescopes—the Planck satellite, the Atacama Cosmology Telescope, and the South Pole Telescope—along with new measurements from the Dark Energy Spectroscopic Instrument. Their goal was to see if the current data could rule out specific models of the early universe, particularly those that predict a slightly different pattern of fluctuations than the standard model suggests. In their initial analysis, using a standard set of assumptions about the universe's contents (a six-parameter model), they found that two popular theories, known as Starobinsky inflation and the Bi-thermal Big Bang model, did not fit the data well. These models were effectively ruled out with a high degree of confidence within that restricted framework.

However, the story changed when the researchers allowed for more flexibility in their calculations. They realized that certain properties of the universe, such as the total number of light, invisible particles known as neutrinos and the sum of their masses, were not perfectly known. When they freed these variables to vary alongside the other parameters (creating a nine-parameter model), the uncertainty in their measurements grew significantly. This broader margin of error meant that the data could no longer definitively exclude the Starobinsky and Bi-thermal models. Instead, these theories became consistent with the observations once again. The researchers found that the current data is simply not precise enough to distinguish between these competing ideas when all the unknowns are taken into account.

Looking ahead, the team used computer simulations to forecast what future telescopes would achieve. They modeled the capabilities of two upcoming projects: the Simons Observatory and a proposed mission called CMB-HD. These future instruments are designed to be far more sensitive and detailed than anything currently in operation. The simulations showed that these next-generation facilities would dramatically sharpen the picture. While current data can measure the strength of the primordial fluctuations to within about half a percent, the future telescopes could tighten this to one-tenth of a percent. More importantly, they would be able to probe much smaller scales of the universe than ever before, reaching distances that are currently unmeasurable.

The results of these future forecasts are striking. If the universe behaves as the current best measurements suggest, the Simons Observatory would be powerful enough to rule out the Bi-thermal Big Bang model and the Starobinsky inflation model with very high confidence. The even more advanced CMB-HD mission would be capable of ruling out all the specific early-universe models the team considered, including those involving complex physics from string theory and supersymmetry, assuming the best-fit values of the spectral index and its running remain as indicated by current data. The researchers also noted that if the current slight preference for a specific type of change in the fluctuation pattern holds true, the CMB-HD mission would decisively eliminate these alternative theories.

Beyond testing specific theories, the study also focused on mapping the primordial power spectrum directly, without assuming it follows a simple mathematical curve. By dividing the universe into different size scales and measuring the fluctuations in each, the team confirmed that the current data aligns perfectly with the standard model of a simple, smooth power law. They found that adding new data from the Dark Energy Spectroscopic Instrument did not significantly change these results, but the inclusion of the South Pole Telescope data did improve the precision. The team released their computer code and updated their tools to help other scientists perform similar analyses, ensuring that the community is ready to interpret the flood of high-quality data that will arrive from the next generation of cosmic experiments. Ultimately, this work highlights that while current observations have narrowed the field, it is the future of high-precision astronomy that will finally reveal the true nature of the universe's birth.

Technical Summary: Current and Future Constraints on the Primordial Power Spectrum

Problem and Motivation
The primordial scalar power spectrum serves as a critical probe of early-universe physics, particularly inflationary scenarios. While many models predict specific deviations from a pure power-law form—characterized by the scalar spectral index ( $n_s$ ), its running ( $\alpha_s$ ), or general shape variations—current constraints are often limited by parameter degeneracies. Specifically, the paper addresses how freeing the effective number of relativistic species ( $N_{\text{eff}}$ ) and the sum of neutrino masses ( $\sum m_\nu$ ) broadens error contours on $n_s$ and $\alpha_s$ , potentially allowing early-universe models previously ruled out by standard six-parameter $\Lambda$ CDM analyses to remain viable. Furthermore, the work seeks to forecast how next-generation Cosmic Microwave Background (CMB) experiments will sharpen these constraints and extend measurements to smaller scales.

Methodology
The authors employ a combination of current observational data and future Fisher matrix forecasts:

Current Datasets (CMB-PAS + DESI): The analysis utilizes a joint dataset comprising:
- CMB-PAS: A combination of Planck 2018 PR3 likelihoods (high- $\ell$ TT, TE, EE; low- $\ell$ TT; and low- $\ell$ EE via SimAll or Sroll2), ACT DR6 CMB-only likelihood, and SPT-3G D1 CMB-only likelihood.
- Lensing: The extended ACT-Planck-SPT (APS) lensing likelihood.
- BAO: Dark Energy Spectroscopic Instrument (DESI) Data Release 1 and 2 (DR1/DR2) measurements.
- Analysis Tools: Parameter constraints are derived using Markov Chain Monte Carlo (MCMC) chains via Cobaya, interfaced with the Einstein-Boltzmann codes CAMB and CLASS. The authors implement higher accuracy settings for CLASS to ensure consistency with CAMB for high-resolution forecasts.
Parameter Space: The study varies standard $\Lambda$ CDM parameters alongside:
- Running of the spectral index ( $\alpha_s$ ).
- Effective number of relativistic species ( $N_{\text{eff}}$ ).
- Sum of neutrino masses ( $\sum m_\nu$ ).
- Dark energy equation of state parameters ( $w_0, w_a$ ) for evolving models.
- A binned primordial power spectrum $P(k)$ , allowing for non-power-law shapes across specific $k$ -bins.
Future Forecasts: Using mock data and covariance matrices simulating the Simons Observatory (SO) and the proposed CMB-HD experiment, the authors perform Fisher matrix analyses. These forecasts include:
- Lensed CMB spectra (TT, TE, EE, BB) and lensing convergence ( $\kappa\kappa$ ).
- Marginalization over baryonic physics parameters (using HMcode2020) and kinetic Sunyaev-Zel'dovich (kSZ) spectrum shape parameters.
- Forecasts for both power-law parameters ( $n_s, \alpha_s$ ) and binned $P(k)$ reconstructions.

Key Contributions

Re-evaluation of Early-Universe Models: The paper demonstrates that in a nine-parameter model ( $\Lambda$ CDM + $\alpha_s$ + $N_{\text{eff}}$ + $\sum m_\nu$ ), the constraints on $n_s$ and $\alpha_s$ broaden significantly compared to the six-parameter $\Lambda$ CDM model. Consequently, models such as Starobinsky inflation and the Bi-thermal Big Bang, which are excluded at $\sim95\%$ confidence in the six-parameter model, become consistent with current data.
Binned Power Spectrum Constraints: The authors provide updated constraints on a binned primordial power spectrum $e^{-2\tau}P(k)$ using the combined CMB-PAS dataset, showing improved precision over previous P-ACT-LB analyses, particularly at $k \approx 0.2 \text{ Mpc}^{-1}$ .
Code and Likelihood Updates: The authors release updated forecast code and public likelihoods (CMB-HD) that support both CAMB and CLASS, ensuring cross-code consistency for high-precision forecasts. They also detail the necessary accuracy settings for CLASS to avoid biases in CMB-HD-scale forecasts.

Results

Current Constraints: Using CMB-PAS + DESI DR2 in the nine-parameter model, the authors find $n_s = 0.9758 \pm 0.0064$ and $\alpha_s = 0.0080 \pm 0.0064$ . These results are consistent with a wide range of inflationary and non-inflationary models that were previously disfavored.
Future Projections:
- SO-like Experiments: Projected to constrain $e^{-2\tau}P(k=0.2 \text{ Mpc}^{-1})$ to $\sim0.1\%$ uncertainty.
- CMB-HD-like Experiments: Projected to tighten the $k=0.2 \text{ Mpc}^{-1}$ constraint to $0.1\%$ and extend measurements to $k=30 \text{ Mpc}^{-1}$ with uncertainties within a factor of ten.
- Model Discrimination: While current data cannot rule out the tested early-universe models in the nine-parameter context, the authors forecast that SO could rule out specific models (e.g., Bi-thermal Big Bang, Starobinsky) at the $3\sigma$ level, and CMB-HD could rule out all considered models by at least $6\sigma$ (up to $17\sigma$ for Starobinsky), assuming current best-fit values hold.
Baryonic Physics Impact: Including baryonic feedback and kSZ marginalization increases the error on $\alpha_s$ by approximately $32\%$ but affects other cosmological parameters by at most $14\%$ .

Significance
The paper establishes that while current data, when analyzed with a broader parameter space including neutrino mass and relativistic species, cannot definitively exclude specific early-universe scenarios, future CMB facilities will provide the necessary precision to do so. The work highlights the critical role of upcoming experiments like SO and CMB-HD in breaking degeneracies between cosmological parameters and primordial power spectrum features. By extending measurements to smaller scales ( $k \sim 30 \text{ Mpc}^{-1}$ ) and achieving sub-percent precision on the power spectrum amplitude, these future surveys will offer a decisive test of inflationary and alternative early-universe models. The authors emphasize that their results are contingent on the assumption that current best-fit values for $n_s$ and $\alpha_s$ remain valid; if $\alpha_s$ remains positive, CMB-HD would decisively rule out models preferring $\alpha_s \approx 0$ .

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