First constraints on causal sources of primordial gravitational waves from BICEP/Keck, SPTpol, SPT-3G, Planck and WMAP $B$-mode data

TL;DR

This work constrains non-inflationary primordial gravitational waves by targeting early causal tensor (ECT) sources that yield a white-noise spectrum on CMB scales, encoded as $\mathcal{P}_h(k) = r_{ect} A_s (k/k_{ref})^3$ with $k_{ref}=0.01\ \mathrm{Mpc}^{-1}$. Using a joint, multi-dataset likelihood from BICEP/Keck, SPTpol, SPT-3G, Planck, and WMAP, the authors marginalize over lensing and foregrounds and perform MCMC with priors on the tensor amplitudes, yielding a 95% CL upper limit of $r_{ect} < 0.0077$ and $r < 0.033$, with a translation to ultra-low-frequency GW energy density $\Omega_{GW} h^2$. The analysis demonstrates that CMB $B$-mode data probe GW backgrounds inaccessible to traditional detectors, and the lensing prior helps break degeneracies between ECT and lensing. These results impose strong constraints on a broad class of post-inflationary GW sources (e.g., phase transitions, scalar-induced GWs, defects) and establish a foundation for future, more selective tests of early-Universe physics using CMB polarization.

Abstract

Non-inflationary sources of gravitational waves in the early Universe generically predict causality-limited tensor power spectra at low frequencies. We report the first-ever constraints on such sources based on cosmic microwave background (CMB) $B$-mode polarization measurements. Using data from BICEP/Keck, SPTpol, SPT-3G, Planck, and WMAP, we constrain the amplitude of an early causal tensor (ECT) power spectrum parameterized by $r_{ect}$, the ratio of causal tensor power to total scalar power at $k~=~0.01$ Mpc$^{-1}$, and obtain a 95% CL upper limit of $r_{ect}<$ 0.0077. Since $r_{ect}$ can easily be related to the parameters of a given theory, our bound robustly constrains a broad class of well-motivated gravitational wave sources in the early universe, including first-order cosmological phase transitions, enhanced small-scale density perturbations, and various topological defects. Finally, we translate our limit into a bound on the present-day energy density in gravitational waves at ultra-low frequencies otherwise inaccessible to traditional gravitational wave detection strategies, including pulsar timing arrays, interferometers, and resonant cavities.

Figures (5)

• Figure 1: Tensor power spectra $\mathcal{P}_h(k)$ for select ECT models presented by TheoryPaper: a first-order cosmological phase transition (red), scalar-induced GWs (green), and cosmic strings (purple). The gray "linear CMB" region marks the domain of support of the window function $\mathcal{F}_\ell(k)$ in Eq. \ref{['eq:DlBB']} for $20 < \ell < 2301$. As the models asymptote to the same $k^3$ scaling required by causality over the CMB regime, all three models predict identical $B$-mode power spectra. For comparison, we show a scale-invariant inflationary spectrum (blue).
• Figure 2: Theoretical $B$-mode power spectra for: inflationary GWs (blue dashed); ECT sources (red solid); and gravitational lensing (gray dot-dashed). The curves correspond to $r = 0.02$, $r_{\rm ect}{}= 0.02$, and $A_{\mathrm{LT}} = 1$.
• Figure 3: Constraints on the tensor-to-scalar ratio $r$, the ratio of causal tensor to scalar power $r_{\rm ect}$, and the amplitude of lensing $B$ modes $A_{\rm LT}$ from a combination of BICEP/Keck, SPTpol, SPT-3G, Planck, and WMAP data. The diagonal panels show the 1D posterior distributions of the parameters and the off-diagonal panels show the 2D 68% and 95% confidence intervals.
• Figure 4: Measured $B$-mode power spectra from SPT-3G (diamonds), SPTpol (empty circles), and BICEP/Keck 2018 (filled circles), plotted against the predicted lensing $B$-mode power with $A_{\mathrm{LT}} = 1.03$ (dash-dot grey line). The plotted SPT-3G balkenhol2024compressed and BICEP/Keck points are CMB-only for visualization purposes; however parameter constraints use the full multi-frequency likelihood with foreground marginalization. We show model lines corresponding to the derived 95% CL upper limits on the ECT (red) and inflationary (blue) GW power. The upper limit on ECT GWs is set by the absence of excess power at high multipoles, above that expected for lensing $B$-modes.
• Figure 5: Limits on the energy density in GWs today, as parameterized by the spectral density $\Omega_{\rm GW}h^2$. The red shaded region labeled "CMB: ECT" is excluded at 95% CL for GWs sourced by ECT sources, with the bolded portion corresponding to frequencies directly probed by the CMB. The dashed red line is a forecast for future ECT constraints from SPT-3G+. The gray shaded region labeled "CMB: Inflation" is excluded at 95% CL for GWs sourced by inflation with a spectrum parameterized by $r$ and $n_t$Lasky:2015. We also show limits (solid shaded) and projected sensitivities (dashed) from various PTAs and GW interferometers Schmitz:2020sylCaldwell:2022qsjNANOGrav:2020Shannon:2015EPTA:2015qepIPTA:2016Weltman:2018zrlCaprini:2019Isoyama:2018rjbYagi:2011wgET:2019dnzReitze:2019ioxKAGRA:2021kbb. Note that the red and gray exclusion curves apply only to specific sources (ECT sources and inflation).