The first observational constraint on non-inflationary causal sources of primordial gravitational waves used cosmic microwave background (CMB) BB-mode polarization data from BICEP/Keck, SPTpol, SPT-3G, Planck and WMAP to bound the amplitude of an 'early causal tensor' (ECT) spectrum, rect<0.0077r_{\rm ect}<0.0077 (95% CL), exploiting the universal white-noise k3k^3 tensor scaling that causality forces for any gravitational-wave source acting before z∼7×104z\sim7\times10^4. That bound applies only to the generic template. Many motivated models — first-order phase transitions, topological defects, scalar-induced tensors — predict spectra that peak and turn over on CMB scales, and sources active at later times evade the template entirely. The open problem is to extend causal-source BB-mode science beyond this single template: to constrain with current data the causal tensor spectra of the specific, cosmologically compelling models that deviate from the universal white-noise k3k^3 form — the peaked/turnover spectra predicted by first-order phase transitions, topological defects and scalar-induced tensors, and sources active at later times or with nonstandard temporal behavior — while controlling the analysis systematics the first analysis deferred (in particular its per-experiment foreground parameterizations) and, where model-specific limits are obtained, translating them into bounds on the present-day gravitational-wave energy density ΩGW(f)\Omega_{\rm GW}(f) at ultra-low frequencies.

Background

A curl-like polarization pattern ("BB modes") in the cosmic microwave background is the characteristic observational signature of primordial gravitational waves. In inflationary cosmology, tensor perturbations stretched from quantum fluctuations predict a nearly scale-invariant spectrum, conventionally parameterized by the tensor-to-scalar ratio rr. In 2024–2026 it was recognized that non-inflationary sources — first-order cosmological phase transitions, networks of topological defects (strings, domain walls), and gravitational waves induced at second order by enhanced small-scale density perturbations — can also produce observable BB modes. For any such source, causality forbids correlations on scales larger than the horizon and finite source duration bounds the correlation length, so the dimensionless tensor power spectrum must behave as white noise at small wavenumber: Ph(k)∝k3\mathcal{P}_h(k)\propto k^3. The spectrum turns over at a wavenumber set by the source's correlation length, but if the source acts before redshift z∼7×104z\sim7\times10^4 this turnover is hidden in the CMB by Silk damping — the exponential damping of small-scale CMB anisotropies caused by photon diffusion, whose scale matches the horizon size at that redshift. Any non-inflationary source generating gravitational waves exclusively before z∼7×104z\sim7\times10^4 therefore predicts a universal BB-mode angular shape (peaking near multipole ℓ∼1000\ell\sim1000, unlike the inflationary peak near ℓ∼100\ell\sim100); such sources are called early causal tensor (ECT) sources, and their amplitude is parameterized by rectr_{\rm ect}, the ratio of causal tensor power to measured scalar power at the reference wavenumber kref=0.01 Mpc−1k_{\rm ref}=0.01\,\mathrm{Mpc}^{-1}. Zebrowski et al. (arXiv:2601.20958, 2026) performed the first such analysis, building a multifrequency likelihood over BICEP/Keck, SPTpol, SPT-3G, Planck and WMAP data across 20<ℓ<230120<\ell<2301 and obtaining rect<0.0077r_{\rm ect}<0.0077 at 95% confidence, which they translate into a bound on the present-day gravitational-wave energy density ΩGW(f)\Omega_{\rm GW}(f) at frequencies f∼10−19f\sim10^{-19} Hz — far below the reach of pulsar timing arrays and interferometers. The analysis has known limitations and adopted choices that frame the present problem: each experiment's likelihood marginalized over its own independent foreground parameters (a combined multifrequency foreground model was explicitly deferred); the shape degeneracy between ECT and lensing BB modes was broken with an external Gaussian prior on the lensing-BB-mode amplitude, ALT∼N(1.013,0.0232)A_{\rm LT}\sim N(1.013,0.023^2) from other lensing measurements, an adopted tool that strengthens the limit on rectr_{\rm ect}; the rect→ΩGW(f)r_{\rm ect}\to\Omega_{\rm GW}(f) translation is model-dependent through the source redshift, tensor transfer functions, and the epoch of horizon re-entry; and the entire bound assumes a non-inflationary source active before Silk damping. A companion theory paper (Greene, Ireland, Krnjaic and Tsai, arXiv:2601.20967, published in Physical Review D 2026) supplies the unified ECT framework, the rectr_{\rm ect} parametrization, and case studies mapping specific models (phase transitions, cosmic strings, scalar-induced tensors) onto it — but the targeted observational analyses within the ECT framework remain to be done; the existing data-driven BB-mode constraint on stable defect networks (reference [3]) is a separate, non-ECT analysis.

Problem Statement

Extend the first CMB BB-mode constraint on causal primordial gravitational-wave sources (Zebrowski et al., arXiv:2601.20958: rect<0.0077r_{\rm ect}<0.0077 at 95% CL on the white-noise k3k^3 early-causal-tensor template) to the broader class of non-inflationary causal sources whose tensor spectra deviate from that template. Following the source paper's outlook, the objective is targeted observational constraints, using current BB-mode data (BICEP/Keck, SPTpol, SPT-3G, Planck, WMAP), on causal tensor power spectra that peak and turn over within the range of CMB scales — as predicted by specific motivated models such as first-order phase transitions, topological defects, and scalar-induced tensors — and on other causal sources whose spectra deviate from the universal white-noise form, such as sources active after z∼7×104z\sim7\times10^4 or with nonstandard temporal behavior. A new data-driven constraint on the tensor spectrum of any one of these enumerated classes resolves this core target, provided the solution identifies the class it constrains, shows why that spectrum deviates from the universal white-noise k3k^3 form, and demonstrates that the result is not already supplied by published data-driven BB-mode analyses of the same source — in particular reference [3], which already constrains stable cosmic string and domain wall networks; the remaining classes are additional scope rather than a requirement. Two further items are natural components of a robust realization of this objective, though neither is prescribed for every solution nor a substitute for the core target: the analysis robustness the source analysis explicitly deferred — a single unified multifrequency foreground model across the combined experiments, replacing the independent per-experiment foreground parameterizations — and, once model-specific limits are obtained, their translation into bounds on the present-day gravitational-wave energy density ΩGW(f)\Omega_{\rm GW}(f), propagating uncertainties in the source redshift, tensor transfer functions, and mode re-entry epochs. The common objective is to convert the single-template proof-of-concept bound into robust, systematics-controlled constraints on early-universe causal gravitational-wave sources beyond the white-noise k3k^3 template. A pure forecast with no new constraint on data does not answer the problem.

References

  1. Jessica A. Zebrowski, Aurora Ireland, Christian L. Reichardt, Kylar Greene, Gordan Krnjaic, Yuhsin Tsai, François R. Bouchet, 'First constraints on causal sources of primordial gravitational waves from BICEP/Keck, SPTpol, SPT-3G, Planck and WMAP BB-mode data' (2026), arXiv:2601.20958, doi:10.48550/arXiv.2601.20958, https://arxiv.org/abs/2601.20958
  2. Kylar Greene, Aurora Ireland, Gordan Krnjaic, Yuhsin Tsai, 'Universal CMB BB-mode spectrum from early causal tensor sources', Physical Review D 113, 123504 (2026), arXiv:2601.20967, doi:10.1103/vzpt-4wqc, https://arxiv.org/abs/2601.20967
  3. Luca Caloni, Ricardo Z. Ferreira, Lara Sousa, Clara Winckler, 'Cosmic strings and domain walls: the impact of CMB BB-mode data' (2026), arXiv:2602.20050, https://arxiv.org/abs/2602.20050
  4. Emilie Hertig, Antón Baleato Lizancos, Frank J. Qu, J. Richard Bond, Erminia Calabrese, Anthony Challinor, Mark J. Devlin, Jo Dunkley, Thibaut Louis, Mathew S. Madhavacheril, Toshiya Namikawa, Lyman A. Page, Neelima Sehgal, Blake Sherwin, Cristóbal Sifón, Suzanne T. Staggs, Edward J. Wollack, 'The Atacama Cosmology Telescope: BB-mode delensing with DR6 data and external tracers of large-scale structure' (2025), arXiv:2511.21949, https://arxiv.org/abs/2511.21949
  5. Kylar Greene, Aurora Ireland, Gordan Krnjaic, Yuhsin Tsai, 'Observable CMB BB-modes from Cosmological Phase Transitions', Physical Review D 113, 083527 (2026), arXiv:2410.23348, doi:10.1103/gv1d-znp4, https://arxiv.org/abs/2410.23348
  6. Aurora Ireland, Kuver Sinha, Tao Xu, 'From Fluctuation to Polarization: Imprints of O(1−10) Mpc−1O(1-10)\,\mathrm{Mpc}^{-1} Curvature Perturbations in CMB BB-modes from Scalar-Induced Gravitational Waves', Journal of Cosmology and Astroparticle Physics 10 (2025) 034, arXiv:2507.02044, doi:10.1088/1475-7516/2025/10/034, https://arxiv.org/abs/2507.02044
  7. Gordan Krnjaic, Duncan Rocha, Huangyu Xiao, 'Primordial Neutron Stars' (2026), arXiv:2604.08651, https://arxiv.org/abs/2604.08651
  8. Daven Wei Ren Ho, Ameen Ismail, Yuhsin Tsai, 'Probing Confining Dark Sectors with Cosmological Perturbations' (2026), arXiv:2606.25014, https://arxiv.org/abs/2606.25014
  9. Kylar Greene, Daven Wei Ren Ho, Soubhik Kumar, Yuhsin Tsai, 'Cosmological and Astrophysical Constraints on Late First-Order Phase Transitions' (2026), arXiv:2603.00272, https://arxiv.org/abs/2603.00272