P-ACT-LB-BK18: CMB, BAO & Inflation Constraints

• P-ACT-LB-BK18 data is an integrated cosmological dataset combining ACT, Planck, SPT, BK18, and DESI BAO observations to rigorously constrain inflation models.
• It employs advanced Bayesian MCMC methods with tailored likelihoods to analyze high-resolution CMB power spectra and extended tensor power spectra.
• The combined data impose stringent upper bounds on the tensor-to-scalar ratio (r ≲ 10⁻³–10⁻²), reshaping the viable inflationary model space by excluding high-r scenarios.

The P-ACT-LB-BK18 data set refers to the integrated collection of cosmological observations from the Atacama Cosmology Telescope (ACT), Planck, South Pole Telescope (SPT), BICEP/Keck (represented as BK18), and the Dark Energy Spectroscopic Instrument (DESI) Baryon Acoustic Oscillation (BAO) surveys. This combination—often denoted as P-ACT-LB-BK18—forms the most stringent suite of current cosmic microwave background (CMB) and large-scale structure data, providing leading constraints on inflationary models, the primordial tensor-to-scalar ratio, and the early universe’s fundamental physics.

1. Dataset Composition and Observational Scope

The P-ACT-LB-BK18 data set comprises several complementary experiments:

• Atacama Cosmology Telescope (ACT DR4, DR6): Delivers high-resolution TT, TE, and EE CMB power spectra over multipoles $350 < \ell < 8000$. Its extended high-$\ell$ coverage offers an independent handle on the scalar spectral index $n_s$ and serves as a cross-check to satellite data.
• Planck: Provides the tightest CMB anisotropy measurements and lensing constraints, enabling robust determination of $n_s$ and $r$, and serving as the baseline for cosmological parameter extraction.
• South Pole Telescope (SPT): Offers extended high-multipole data, valuable for cross-comparison.
• BK18 (BICEP2/Keck/BICEP3 through 2018): Supplies targeted measurements of the CMB $B$-mode polarization at low multipoles, directly probing the imprint of primordial gravitational waves (PGWs).
• DESI BAO: Constrains the expansion history and helps break parameter degeneracies via low-$z$ structure formation.

The integration of these data sets is essential for both direct (e.g., $B$-mode detection) and indirect (e.g., expansion history via BAO) probes of early universe physics.

2. Theoretical Frameworks and Key Quantities

The combined data set enables analyses that probe the tensor sector of primordial scalar and tensor fluctuations by employing an extended tensor power spectrum: $$\mathcal{P}_T(k) = A_t \left(\frac{k}{k_*}\right)^{n_t + \frac{1}{2} n_{trun} \ln(k/k_*)},$$ where $A_t$ is the tensor amplitude, $n_t$ the tensor spectral tilt, $n_{trun}$ its running, and $k_*$ the tensor pivot scale ($0.05\,\mathrm{Mpc}^{-1}$ is typical). The tensor-to-scalar ratio is defined as $r = A_t/A_s$.

The analysis exploits next-order inflationary consistency relations: $$n_t = -\frac{r}{8} \left(2 - \frac{r}{8} - n_s\right), \qquad n_{trun} = \frac{r}{8}( \frac{r}{8} + n_s - 1).$$ This framework links CMB observables to inflationary potential dynamics, enabling precision testing of competing scenarios.

3. Analysis Methodology

The P-ACT-LB-BK18 studies employ a Bayesian parameter estimation framework. The key aspects include:

• MCMC and Likelihood Sampling: Theoretical $C_\ell$ spectra are computed (with CAMB), and the full parameter space is explored via MCMC (with CosmoMC).
• Logarithmic Prior on $r$: A prior uniform in $\log_{10}r \in [-30, 5]$ is adopted to account for the possibility that $r$ could span many orders of magnitude, in contrast to previous uniform priors.
• Combined Data Likelihoods: Likelihoods are constructed for Planck, ACT, and SPT individually and then in joint combinations with BK18 and DESI BAO (e.g., BAD = BK18+ACT+DESI).

Upper bounds on $r$ and posterior shapes are thereby conditioned on this comprehensive data set.

4. Constraint Results and Comparative Performance

The P-ACT-LB-BK18 combination yields the most stringent current constraints on primordial gravitational waves:

Data Set	Upper Limit $r$ (2$\sigma$)	$n_s$ Constraint
Planck only	$r < 0.0037$	tightest, central $n_s$
ACT only	$r < 0.2274$	complementary, prefers $n_s \approx 1$
SPT only	$r < 0.5026$	less constraining, agrees on $n_s$
ACT+BK18+DESI	strong reduction in $r$	reshaped $(n_s, \log_{10} r)$ parameter space

• Planck: The gold standard for $r$ constraints, compressing viable inflationary model space.
• ACT & SPT: Albeit less constraining on $r$, they are vital for high-$\ell$ cross-checks, with ACT preferring a nearly scale-invariant spectrum.
• BK18: Essential for $B$-mode ($r$) posterior with a sharp peak near $r \sim 10^{-2}$, though lacking a lower bound.
• DESI BAO: Tightens cosmological parameters by breaking degeneracies and providing independent expansion rate constraints.

Combined, these datasets allow the exclusion of a significant portion of the inflationary model space, especially those with large $r$.

5. Impact on Inflationary Models

The joint P-ACT-LB-BK18 constraints have significant ramifications:

• Models Excluded: Natural inflation and concave-potential models are ruled out at $\gtrsim 2\sigma$; even $R^2$ (Starobinsky) inflation faces tension.
• Parameter Shifts: Severe restriction in $r$ (to as low as $10^{-3}$–$10^{-2}$) demands re-evaluation of inflationary model viability.
• Phenomenology: Only scenarios able to accommodate $r \lesssim 10^{-3}$—including tuned plateau and attractor models—remain consistent.

This suggests that single-field slow-roll inflationary models with large tensor modes are increasingly disfavored, motivating the search for architectures that yield very small $r$ without significant fine-tuning.

6. Implications for Cosmic Origins and Fundamental Physics

The P-ACT-LB-BK18 data set:

• Probes Inflation Energy Scales: $\sim 10^{16} \,\mathrm{GeV}$ becomes a secure upper limit on energy scales of inflation.
• Tests Quantum Gravity: The absence of PGWs at current sensitivity precludes direct evidence for quantized gravitons, but further tightening could provide sweeping constraints on quantum gravitational imprints.
• Influences Experimental Design: Motivates next-generation CMB and $B$-mode experiments (e.g., CMB-S4, LiteBIRD) to reach sensitivity to $r \sim 10^{-3}$ and below.

A plausible implication is that incremental advances in $B$-mode sensitivity may yield diminishing returns for conventional inflation, but could decisively confirm or preclude certain classes of theory.

7. Synthesis and Perspective

The P-ACT-LB-BK18 dataset represents the state-of-the-art in constraining early universe cosmology through multi-band, multi-experiment CMB and large-scale structure observations. The primary outcomes include:

• Restriction of viable inflationary models to those with low $r$ and compatible $n_s$ (notably plateau/attractor types).
• Exclusion of a wide class of high-$r$ models at $\sim 2\sigma$.
• Establishing the role of combined $B$-mode, high-$\ell$ anisotropy, and BAO data as essential for breaking parameter degeneracies.

This comprehensive, multi-instrument approach exemplifies the precision cosmological methodology required for robust progress in inflationary model discrimination and the ongoing effort to connect CMB observations to fundamental physics at ultrahigh energy scales.