Cross-frequency SGWB anisotropy from compact topology: CMB B-mode covariance as a transfer probe
Abstract
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 δCBBℓ m,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 TB/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 Lqmax=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.
Versions
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BibTeX
@misc{qiang2026crossfrequency,
title = {{Cross-frequency SGWB anisotropy from compact topology: CMB B-mode covariance as a transfer probe}},
author = {Li-E Qiang and Peng Xu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.09697},
eprint = {2608.09697},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09697},
}
RIS
TY - UNPB
TI - Cross-frequency SGWB anisotropy from compact topology: CMB B-mode covariance as a transfer probe
AU - Li-E Qiang
AU - Peng Xu
PY - 2026
DA - 2026/08/10
PB - arXiv
DO - 10.48550/arxiv.2608.09697
UR - https://arxiv.org/abs/2608.09697
AB - 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 q_LM(k), we obtain an explicit kernel that maps q_LM(k) onto the off-diagonal covariance δC^BB_ℓ m,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 TB/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 q_LM--Gaunt contractions agree at L_q^max=2ℓ_max 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.
ER -