Planck PR4 and BICEP/Keck 2018 tensor-to-scalar upper limit

The reported parameter is the source-defined tensor-to-scalar amplitude ratio $r:=A_t\mathrm{/}{A}_s$: $A_t$ and $A_s$ are the tensor and scalar primordial power-law amplitudes used by the article's CAMB/Cobaya model at its pivot. The local spectrum convention in Primordial scalar and tensor observables for inflation models defines a ratio ${\mathcal{P}}_t\mathrm{/}{\mathcal{P}}_{\zeta }$ with both tensor polarizations included. Identifying that local ratio with the article's $r$ additionally requires matching its scalar curvature variable and tensor-polarization normalization; §II of the cited article does not state that variable map. The upper limits recorded here therefore remain limits on the article's $A_t\mathrm{/}{A}_s$, without asserting that additional identification. Tristram et al. analyze a flat $\mathrm{\Lambda }$CDM+$r$ model with the six base scalar parameters and $r$. Scalar and tensor primordial spectra are power laws with no running; the tensor spectral index is fixed by the single-field slow-roll consistency relation. They evaluate $r$ at $k_{\ast }=0.05{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$. This is the paper's conventional comoving inverse-megaparsec pivot; its numerical value is retained in that source coordinate convention.

The analyzed BICEP/Keck likelihood is formed from BICEP2, Keck Array and BICEP3 observations through the 2018 season. The paper describes joint BB auto- and cross-spectra at 95, 150 and 220 GHz, together with WMAP 23/33 GHz and Planck PR4 30, 44, 143, 217 and 353 GHz maps over an effective area of about 400 square degrees (about 1% of the sky), selected for low foreground emission. Its data model includes Galactic dust and synchrotron emission and their correlations. The Planck low-multipole likelihood uses component- separated maps over 50% of the sky. The analysis neglects cross-correlations between the BICEP/Keck and Planck data sets, citing their different sky coverage; this is an assumption of the reported likelihood combination, not a reanalysis performed here.

Primordial scalar and tensor observables for inflation models

Tristram et al., §§II–IV

The analysis combines publicly available likelihoods using the CAMB Boltzmann solver and Cobaya Markov-chain Monte Carlo sampling. It uses the BICEP/Keck likelihood through the 2018 observing season, Planck PR4 low-multipole polarization and high-multipole TT, TE and EE data, then considers BAO and CMB lensing as additional data. The Planck low-multipole polarization likelihood uses cross-spectra and marginalizes over uncertainty in the covariance estimated from 400 PR4 simulations. The high-multipole Planck spectra use the 100, 143 and 217 GHz detector-set maps. The source models foreground residuals and retains covariance terms rather than setting all off-diagonal entries to zero.

Calibration here means estimating the multiplicative response that converts a detector reading to a temperature fluctuation in thermodynamic CMB kelvin. For the PR4 maps, NPIPE fits the seasonal CMB temperature-dipole template produced by the spacecraft's motion around the Sun, including corrections for response outside the main beam. Section 6.1 tests gain recovery by regressing the injected dipole against simulated maps after subtraction of the input CMB and foregrounds and masking the more contaminated half of the sky. Table 7 reports frequency-dependent gain scatter and bias. Its calibration also filters some large-scale polarization; §4.3 measures that response using simulations. The low-multipole likelihood described by Tristram et al. propagates the instrumental effects represented in its PR4 simulations, rather than treating these maps as free of calibration error.

BICEP2 §13.3 and BICEP3 §IX.3 describe relative gains from small telescope elevation changes: the changing atmospheric signal supplies a common response for comparing detectors. Absolute gains use cross-spectra of the instrument's temperature map with externally calibrated CMB maps, with matching beam and filter processing. The BICEP3 gain is averaged over the first five analysis bandpower bins. Its §VIII.9 measures detector polarization response with a rotating polarized source and removes a global angle offset by minimizing EB and TB correlations. The BK18 Appendix F.3 measures BICEP3 and Keck frequency responses with a Fourier-transform spectrometer; a bandpass is this response as a function of frequency, and a band-centre error shifts its assumed frequency scale. These are reported calibration methods and checks; this item does not reconstruct maps, calibration fits, likelihood code or chains.

Tristram et al., §§III–VI and Appendix

Planck Collaboration, NPIPE §§2.4.3, 4.3 and 6.1, Table 7

BICEP2 Collaboration, §13.3

BICEP/Keck XV, §§VIII.9 and IX.3

BICEP/Keck XIII, Appendix F.3

For Planck PR4 plus BICEP/Keck 2018, the authors report $r<0.034$ at 95% confidence. Adding BAO and CMB lensing tightens their reported upper limit to

They report the combined results as consistent with $r=0$. These are likelihood- and model-dependent upper limits, not a detection of primordial tensors.

Tristram et al., Eq. (7), §VI and conclusion

The upper bound inherits the selected likelihood combination, flat $\mathrm{\Lambda }$CDM+$r$ parameterization, fixed tensor-tilt consistency relation, no-running spectral assumption and chain-based posterior analysis. The article labels the reported bound a 95% confidence limit, but does not state the prior on $r$ or define a frequentist coverage calculation for that bound. It is therefore retained here as the source-reported limit, without inferring calibrated frequentist coverage or a posterior probability that a model is true. The Planck low-$\mathrm{\ell }$ polarization likelihood accounts for uncertainty in the estimated covariance using 400 PR4 simulations; the paper estimates the finite-simulation uncertainty in that covariance at about 5% and marginalizes over it. No raw sky maps, full likelihood reconstruction or independent covariance analysis is included here; no precision beyond the reported limit is inferred.

Calibration uncertainties remain conditional on the instrument models and simulation inputs. NPIPE's Table 7 gives gain recovery, not an uncertainty on $r$; its §4.3 also cautions that its simulations have insufficient CMB B-mode signal to measure a B-mode transfer function. BICEP3's polarization-angle and cross-polar response calibration has its own systematic limitations. BK18 Appendix F.3 tests conservative band-centre shifts of $\pm 2\mathrm{\%}$ in simulations and separately marginalizes over per-band fractional shifts with Gaussian mean zero and standard deviation $0.02$. Those tests concern the BK18 baseline analysis; they are not a newly computed error budget for the Tristram PR4 combination. This record neither adds their errors in quadrature to the reported limit nor asserts that every calibration uncertainty was independently audited.

Tristram et al., §§III–VI and Appendix

Planck Collaboration, NPIPE §§4.3 and 6.1, Table 7

BICEP/Keck XV, §VIII.9

BICEP/Keck XIII, Appendix F.3

This analysis constrains $r$ for the stated primordial-spectrum convention and data/model combination. It neither proves an inflationary model nor measures string theory directly. A 95% confidence upper bound is not the probability that a model is true, and the reported compatibility with $r=0$ does not establish that the tensor spectrum vanishes.

Primordial scalar and tensor observables for inflation models

Tristram et al., §VI and conclusion