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Updated Constraints and Forecasts on Primordial Tensor Modes

Giovanni Cabass, Luca Pagano, Laura Salvati, Martina Gerbino, Elena Giusarma, Alessandro Melchiorri

TL;DR

This work sharpens constraints on the primordial tensor power spectrum by combining Planck, BKP, and Keck data, including the GW contribution to $N_{\rm eff}$ and exploring its impact on $r$ and $n_{\rm t}$. It demonstrates that external probes like FIRAS, LIGO-Virgo, and pulsar timing complement CMB data, and shows that a COrE-like mission with delensing could measure $r$ at the few-percent level and constrain $n_{\rm t}$ with ~0.1 precision, though testing the single-field consistency relation remains challenging. The analysis highlights the sensitivity of the tilt constraints to priors on $r$ in the absence of a detection and discusses how reheating and UV cutoffs influence $N^{\rm GW}_{\rm eff}$, underscoring the importance of multi-channel observations for robust inflationary inferences. Overall, the paper maps the current landscape of tensor constraints and outlines realistic paths for significantly tighter future bounds through a combination of CMB polarization, spectral distortions, and direct GW measurements.

Abstract

We present new, tight, constraints on the cosmological background of gravitational waves (GWs) using the latest measurements of CMB temperature and polarization anisotropies provided by the Planck, BICEP2 and Keck Array experiments. These constraints are further improved when the GW contribution $N^{\rm GW}_{\rm eff}$ to the effective number of relativistic degrees of freedom $N_{\rm eff}$ is also considered. Parametrizing the tensor spectrum as a power law with tensor-to-scalar ratio $r$, tilt $n_\mathrm{t}$ and pivot $0.01\,\mathrm{Mpc}^{-1}$, and assuming a minimum value of $r=0.001$, we find $r < 0.089$, $n_\mathrm{t} = 1.7^{+2.1}_{-2.0}$ ($95\%\,\mathrm{CL}$, no $N^{\rm GW}_{\rm eff}$) and $r < 0.082$, $n_\mathrm{t} = -0.05^{+0.58}_{-0.87}$ ($95\%\,\mathrm{CL}$, with $N^{\rm GW}_{\rm eff}$). When the recently released $95\,\mathrm{GHz}$ data from Keck Array are added to the analysis, the constraints on $r$ are improved to $r < 0.067$ ($95\%\,\mathrm{CL}$, no $N^{\rm GW}_{\rm eff}$), $r < 0.061$ ($95\%\,\mathrm{CL}$, with $N^{\rm GW}_{\rm eff}$). We discuss the limits coming from direct detection experiments such as LIGO-Virgo, pulsar timing (European Pulsar Timing Array) and CMB spectral distortions (FIRAS). Finally, we show future constraints achievable from a COrE-like mission: if the tensor-to-scalar ratio is of order $10^{-2}$ and the inflationary consistency relation $n_\mathrm{t} = -r/8$ holds, COrE will be able to constrain $n_\mathrm{t}$ with an error of $0.16$ at $95\%\,\mathrm{CL}$. In the case that lensing $B$-modes can be subtracted to $10\%$ of their power, a feasible goal for COrE, these limits will be improved to $0.11$ at $95\%\,\mathrm{CL}$.

Updated Constraints and Forecasts on Primordial Tensor Modes

TL;DR

This work sharpens constraints on the primordial tensor power spectrum by combining Planck, BKP, and Keck data, including the GW contribution to and exploring its impact on and . It demonstrates that external probes like FIRAS, LIGO-Virgo, and pulsar timing complement CMB data, and shows that a COrE-like mission with delensing could measure at the few-percent level and constrain with ~0.1 precision, though testing the single-field consistency relation remains challenging. The analysis highlights the sensitivity of the tilt constraints to priors on in the absence of a detection and discusses how reheating and UV cutoffs influence , underscoring the importance of multi-channel observations for robust inflationary inferences. Overall, the paper maps the current landscape of tensor constraints and outlines realistic paths for significantly tighter future bounds through a combination of CMB polarization, spectral distortions, and direct GW measurements.

Abstract

We present new, tight, constraints on the cosmological background of gravitational waves (GWs) using the latest measurements of CMB temperature and polarization anisotropies provided by the Planck, BICEP2 and Keck Array experiments. These constraints are further improved when the GW contribution to the effective number of relativistic degrees of freedom is also considered. Parametrizing the tensor spectrum as a power law with tensor-to-scalar ratio , tilt and pivot , and assuming a minimum value of , we find , (, no ) and , (, with ). When the recently released data from Keck Array are added to the analysis, the constraints on are improved to (, no ), (, with ). We discuss the limits coming from direct detection experiments such as LIGO-Virgo, pulsar timing (European Pulsar Timing Array) and CMB spectral distortions (FIRAS). Finally, we show future constraints achievable from a COrE-like mission: if the tensor-to-scalar ratio is of order and the inflationary consistency relation holds, COrE will be able to constrain with an error of at . In the case that lensing -modes can be subtracted to of their power, a feasible goal for COrE, these limits will be improved to at .

Paper Structure

This paper contains 19 sections, 37 equations, 6 figures, 6 tables.

Table of Contents

  1. Introduction
  2. GW spectrum and observations
  3. Nucleosynthesis and primordial abundances
  4. CMB distortions
  5. Pulsar timing + ground- and space-based interferometers
  6. Pulsar timing
  7. Ground- and space-based interferometers
  8. Method
  9. Monte Carlo method and datasets
  10. Simulated datasets for forecasts
  11. Delensing
  12. Results
  13. Current data
  14. Forecasts
  15. Conclusions
  16. ...and 4 more sections

Figures (6)

  • Figure 1: Cartoon plot of a power-law (blue) primordial tensor spectrum ($\Delta^2_\textup{t} = d-10$, $n_{\rm t} = 0.35$) over a range of frequencies $f$ going from $10^{-17}\,\mathrm{Hz}$ (smallest frequency that can be probed with CMB anisotropies) to $10^4\,\mathrm{Hz}$ (largest frequency available to ground-based experiments). The vertical arrows represent the current upper bounds on the tensor amplitude at different scales. We show with a gray dotted line the prior on $r$ used in our analysis ($r > 0.001$). The filled regions show which is the relevant $\Delta f$ for the various observables discussed in the text. We refer to Sec. \ref{['sec:observables-nucleosynthesis']} for a more accurate discussion regarding primordial abundances and the effect on CMB anisotropies. We stress that this plot has only illustrative purposes.
  • Figure 2: Tensor and lensing $B$-modes, together with noise bias for the COrE experiment. The blue spectra correspond $r = 1$, $0.1$, $0.01$ and $n_{\rm t} = 0$, while the orange ones are the lensing $B$-mode spectrum with and without a rescaling by a factor of 0.1. We see that in the case of a $10\%$ delensing the $C^\textup{lens}_\ell$ go completely below noise level.
  • Figure 3: Two-dimensional posterior distributions for $r$ and $n_{\rm t}$: the contours in the left (right) panel are obtained without (with) the inclusion of $N^{\rm GW}_{\rm eff}$. In both panels the red contour is the result for the "vanilla" $\Lambda\text{CDM} + r + n_{\rm t}$ model, using the Planck + BKP dataset. The corresponding $95\%\,\mathrm{CL}$ results for $r$ and $n_{\rm t}$ are reported in Tab. \ref{['tab:results-current']}.
  • Figure 4: Two-dimensional posterior distributions for $r$ and $n_{\rm t}$, using BK14 polarization data: the contours in the left (right) panel are obtained without (with) the inclusion of $N^{\rm GW}_{\rm eff}$. The corresponding $95\%\,\mathrm{CL}$ results for $r$ and $n_{\rm t}$ are reported in Tab. \ref{['tab:results-current-BK']}.
  • Figure 5: Forecasts for $r$ and $n_{\rm t}$ with two different fiducials: $r = 0.05$ (left panel) and $r = 0.01$ (right panel). In both cases the inflationary consistency relation $n_{\rm t} = -r/8$ has been assumed. The corresponding $95\%\,\mathrm{CL}$ limits for $r$ and $n_{\rm t}$ are reported in Tab. \ref{['tab:forecasts-1']}.
  • ...and 1 more figures