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The formula gives $\\delta^{\\rm UH}_{H c}=\\sin^{2}[\\pi \\sqrt{w}/(1+3w)]$ and $\\tilde{\\delta}_{c}=[3(1+w)/(5+3w)]\\sin^{2}[\\pi\\sqrt{w}/(1+3w)]$, where $\\delta^{\\rm UH}_{H c}$ and $\\tilde{\\delta}_{c}$ are the amplitude of the density perturbation at the horizon crossing time in the uniform Hubble slice and the amplitude measure used in numeric"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1309.4201","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"astro-ph.CO","submitted_at":"2013-09-17T07:03:30Z","cross_cats_sorted":["gr-qc","hep-ph","hep-th"],"title_canon_sha256":"66bc163421ae0575c65aa07198da04c6c22d0bfc2101c0f0107e90573a3354dc","abstract_canon_sha256":"9ec05eb2670af30d0d2d0402ae33ed88743eaaeb32cde198ec08c81f31ec56bc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:47:36.836968Z","signature_b64":"hr3CjWbI7FNpdY1A5TaoBE5y9/ldDCCAQY23nYW5qENtYh4UuvYrcZ8/iVQ3xR1v+QyFackVhVBaJRQAvQ6JBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63d7dcc7c2ab83427e5b8508bc6a8cdb935c5f89a68437e2a8e9253ff0fe2ee9","last_reissued_at":"2026-05-18T01:47:36.836298Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:47:36.836298Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Threshold of primordial black hole formation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["gr-qc","hep-ph","hep-th"],"primary_cat":"astro-ph.CO","authors_text":"Chul-Moon Yoo, Kazunori Kohri, Tomohiro Harada","submitted_at":"2013-09-17T07:03:30Z","abstract_excerpt":"Based on a physical argument, we derive a new analytic formula for the amplitude of density perturbation at the threshold of primordial black hole formation in the universe dominated by a perfect fluid with the equation of state $p=w\\rho c^{2}$ for $w\\ge 0$. 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