{
  "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 $F(k,\\hat k)=1+Q(k,\\hat k)$ and its nonmonopole moments as $q_{LM}(k)$, we obtain an explicit kernel that maps $q_{LM}(k)$ onto the off-diagonal covariance $δC^{BB}_{\\ell m,\\ell' m'}$. The kernel factorizes into tensor transfer functions and a spin-weighted Gaunt coefficient and obeys the parity rule $L+\\ell+\\ell'$ 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 $L_q^{\\max}=2\\ell_{\\max}$ to relative Frobenius residuals of $1.4\\times10^{-14}$--$3.0\\times10^{-14}$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Phys. Rev. D - 14 September, 2026",
  "arxiv": "2608.09697",
  "author_total": 2,
  "authors": [
    {
      "name": "Li-E Qiang"
    },
    {
      "name": "Peng Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/30fydgnst0k73.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.09697",
  "fulltext_url": null,
  "id": "30fydgnst0k73",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.09697",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "astro-ph.CO"
    ],
    [
      "arxiv",
      "astro-ph.HE"
    ],
    [
      "arxiv",
      "gr-qc"
    ],
    [
      "openalex",
      "3103|Cosmology and Gravitation Theories"
    ]
  ],
  "openalex": "W7202218285",
  "pdf_url": "https://arxiv.org/pdf/2608.09697",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/30fydgnst0k73.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:19Z",
      "native_id": "2608.09697",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.09697"
    }
  ],
  "subjects": [
    {
      "field": "astronomy",
      "origin": "arxiv:astro-ph.CO",
      "subfield": "cosmology"
    },
    {
      "field": "astronomy",
      "origin": "arxiv:astro-ph.HE",
      "subfield": "high-energy-astrophysics"
    },
    {
      "field": "physics",
      "origin": "arxiv:gr-qc",
      "subfield": "gravitation"
    }
  ],
  "title": "Cross-frequency SGWB anisotropy from compact topology: CMB B-mode covariance as a transfer probe",
  "title_text": "Cross-frequency SGWB anisotropy from compact topology: CMB B-mode covariance as a transfer probe",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:19Z",
  "url": "https://research.draconitic.com/paper/30fydgnst0k73",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1103/ynqz-ykc9",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}