Cross-frequency SGWB anisotropy from compact topology: CMB B-mode covariance as a transfer probe

Compact spatial topology restricts the eigenmodes of primordial tensor perturbations, and the resulting discreteness can render the primordial stochastic gravitational-wave background (SGWB) anisotropic. Here we treat the CMB tensor B-mode covariance as a transfer-filtered measurement of that ultra-low-frequency anisotropy. Writing the normalized angular tensor-power measure as F (k, k̂) = 1 + Q(k, k̂) and its nonmonopole moments as qLM (k), we obtain an explicit kernel that maps qLM (k) onto the off-diagonal covariance δCBB lm,l′ m′. The kernel factorizes into tensor transfer functions and a spin-weighted Gaunt coefficient and obeys the parity rule L + l + l′ even for BB and odd for T B/EB. It is an exact source–response representation of the full compact covariance rather than an additional observable. For a cubic three-torus the geometry pins down a common cubic angular subspace and orientation across frequency bands, although the amplitudes of the allowed multipoles still depend on the radial shell and source spectrum. The same topology-restricted template can therefore be read out either through the CMB B-mode kernel or through the anisotropy response of PTA/LISA/Taiji/TianQin searches. Using CAMB transfer functions and an invariant anisotropic-template statistic, we contrast this tensor channel with the scalar T /E covariance. Independent direct angular-shell sums and qLM –Gaunt contractions agree at Lmax q = 2lmax to relative Frobenius residuals of 1.4 × 10−14 –3.0 × 10−14. The scalar sector holds most of the practical CMB topology information; a fixed-template scan places the combined full-sky S/N = 1 transition between L/χ∗ = 2.34 and 2.36, while the B-mode channel remains subthreshold but isolates the primordial SGWB contribution. We use these results to set out a cross-frequency template for future topology searches.

Improvements to AI systems:

1. Exact kernel-based covariance modeling for non-Euclidean topologies – Implement the derived factorization (tensor transfer functions × spin-weighted Gaunt coefficient) as a differentiable layer in AI pipelines. This enables neural networks or Bayesian inference engines to directly predict off-diagonal CMB B-mode covariance from topological parameters (e.g., torus side lengths), bypassing expensive brute-force simulations.

2. Cross-frequency template matching with parity constraints – Train an AI system to recognize topology signatures across disparate observatories (CMB, PTA, LISA, Taiji, TianQin) by embedding the parity rule (L+l+l′ even/odd) and the common cubic angular subspace into a multi-modal transformer. The system can then fuse data from different frequency bands to produce a unified topology likelihood, improving detection sensitivity for subthreshold signals.

3. Ultra-high-precision numerical validation for AI-generated forecasts – Use the reported Frobenius residuals (1.4×10−14–3.0×10−14) as a benchmark to calibrate AI solvers. An improved AI system can automatically verify its own covariance calculations against this analytic kernel, flagging numerical drift or model misspecification in real time—critical for reliable signal extraction from noisy CMB data.

4. Anisotropy-source separation via transfer-filtered decomposition – Build an AI that decomposes observed angular power into monopole (F=1+Q) and nonmonopole moments (qLM) using the explicit kernel. This allows the system to isolate the primordial SGWB contribution from scalar T/E contamination, even when the B-mode channel is subthreshold, by exploiting the kernel’s source–response structure.

5. Scalar-vs-tensor channel prioritization for survey design – An AI system can learn the optimal observing strategy (sky coverage, frequency bands, integration time) by using the paper’s finding that scalar sector holds most practical topology information, while B-mode isolates SGWB. The system can recommend when to allocate resources to B-mode searches based on predicted S/N transitions (L/χ* = 2.34–2.36).

What the improved AI system can do:

• Given a hypothesized compact topology (e.g., cubic three-torus), it can instantly compute the full CMB B-mode covariance matrix and its off-diagonal structure without Monte Carlo, enabling fast parameter estimation and model comparison.

• It can jointly analyze CMB, PTA, and future space-based interferometer data to search for the same topological template, automatically enforcing parity and angular-subspace consistency across all channels.

• It can detect and correct numerical errors in its own covariance computations to 10−14 precision, making it suitable for next-generation CMB experiments like CMB-S4 or LiteBIRD.

• It can separate primordial gravitational-wave anisotropy from scalar density perturbations in the CMB, providing a clean probe of the early universe even when the B-mode signal is too weak for direct detection.

• It can optimize future observational campaigns by predicting which frequency bands and sky fractions maximize the S/N for a given topological model, balancing scalar and tensor channels.