{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:N23KCZ42XE7AYM2JU2OO3A7G3O","short_pith_number":"pith:N23KCZ42","canonical_record":{"source":{"id":"0809.3036","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2008-09-18T17:01:27Z","cross_cats_sorted":["astro-ph"],"title_canon_sha256":"94c5450707254b17bede0a1c18388eaa949ea15ddab40d6e983fea5c79269e49","abstract_canon_sha256":"215e9f3033adf6e25d71d113d182f99ba183fcbfdfbe0b3b2c1d9c5196a38d99"},"schema_version":"1.0"},"canonical_sha256":"6eb6a1679ab93e0c3349a69ced83e6dbb1adb7bd5b0ccf144bf58f7fdc434194","source":{"kind":"arxiv","id":"0809.3036","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0809.3036","created_at":"2026-05-18T00:12:16Z"},{"alias_kind":"arxiv_version","alias_value":"0809.3036v4","created_at":"2026-05-18T00:12:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0809.3036","created_at":"2026-05-18T00:12:16Z"},{"alias_kind":"pith_short_12","alias_value":"N23KCZ42XE7A","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"N23KCZ42XE7AYM2J","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"N23KCZ42","created_at":"2026-05-18T12:25:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:N23KCZ42XE7AYM2JU2OO3A7G3O","target":"record","payload":{"canonical_record":{"source":{"id":"0809.3036","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2008-09-18T17:01:27Z","cross_cats_sorted":["astro-ph"],"title_canon_sha256":"94c5450707254b17bede0a1c18388eaa949ea15ddab40d6e983fea5c79269e49","abstract_canon_sha256":"215e9f3033adf6e25d71d113d182f99ba183fcbfdfbe0b3b2c1d9c5196a38d99"},"schema_version":"1.0"},"canonical_sha256":"6eb6a1679ab93e0c3349a69ced83e6dbb1adb7bd5b0ccf144bf58f7fdc434194","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:16.388703Z","signature_b64":"mWRwCEGA8dGYtC4+WT4YVstcfp5irDeIKrlC7cIiUDUfuOZuQRa+BSnGcxWzPWMDgeGEXXjbKhjndoUD8TweCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6eb6a1679ab93e0c3349a69ced83e6dbb1adb7bd5b0ccf144bf58f7fdc434194","last_reissued_at":"2026-05-18T00:12:16.388199Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:16.388199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0809.3036","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:12:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4y1vXGGPZmu0AYICJ3yuiZGHQR1hWnklUUL+0fQwpWxPdYgy5JnxxIbczEO4efKTJItJ+Y+6uVn4fttWCoruCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T08:30:03.435642Z"},"content_sha256":"504d31f10794d93107faa46e8a6f00fc4f2c1a1a27ab3d82ea80f03bffe30330","schema_version":"1.0","event_id":"sha256:504d31f10794d93107faa46e8a6f00fc4f2c1a1a27ab3d82ea80f03bffe30330"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:N23KCZ42XE7AYM2JU2OO3A7G3O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The radiative transfer at second order: a full treatment of the Boltzmann equation with polarization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["astro-ph"],"primary_cat":"gr-qc","authors_text":"Cyril Pitrou","submitted_at":"2008-09-18T17:01:27Z","abstract_excerpt":"This article investigates the full Boltzmann equation up to second order in the cosmological perturbations. Describing the distribution of polarized radiation by a tensor valued distribution function, we study the gauge dependence of the distribution function and summarize the construction of the gauge-invariant distribution function. The Liouville operator which describes the free streaming of electrons, and the collision term which describes the scattering of photons on free electrons are computed up to second order. Finally, the remaining dependence in the direction of the photon momentum i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0809.3036","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:12:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eYg6mszwlwd6wUSjExYXNAyNWSCQEThc24Y/AI9k0XXx2zBOMj5WYqboYiMJ2X0235bjdm0cNC3+FkGxioPqCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T08:30:03.435987Z"},"content_sha256":"a8bb8c3cb42d42aeba4cf28c4f32a360b781db9d1078073e862fe762738cb549","schema_version":"1.0","event_id":"sha256:a8bb8c3cb42d42aeba4cf28c4f32a360b781db9d1078073e862fe762738cb549"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N23KCZ42XE7AYM2JU2OO3A7G3O/bundle.json","state_url":"https://pith.science/pith/N23KCZ42XE7AYM2JU2OO3A7G3O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N23KCZ42XE7AYM2JU2OO3A7G3O/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T08:30:03Z","links":{"resolver":"https://pith.science/pith/N23KCZ42XE7AYM2JU2OO3A7G3O","bundle":"https://pith.science/pith/N23KCZ42XE7AYM2JU2OO3A7G3O/bundle.json","state":"https://pith.science/pith/N23KCZ42XE7AYM2JU2OO3A7G3O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N23KCZ42XE7AYM2JU2OO3A7G3O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:N23KCZ42XE7AYM2JU2OO3A7G3O","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"215e9f3033adf6e25d71d113d182f99ba183fcbfdfbe0b3b2c1d9c5196a38d99","cross_cats_sorted":["astro-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2008-09-18T17:01:27Z","title_canon_sha256":"94c5450707254b17bede0a1c18388eaa949ea15ddab40d6e983fea5c79269e49"},"schema_version":"1.0","source":{"id":"0809.3036","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0809.3036","created_at":"2026-05-18T00:12:16Z"},{"alias_kind":"arxiv_version","alias_value":"0809.3036v4","created_at":"2026-05-18T00:12:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0809.3036","created_at":"2026-05-18T00:12:16Z"},{"alias_kind":"pith_short_12","alias_value":"N23KCZ42XE7A","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"N23KCZ42XE7AYM2J","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"N23KCZ42","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:a8bb8c3cb42d42aeba4cf28c4f32a360b781db9d1078073e862fe762738cb549","target":"graph","created_at":"2026-05-18T00:12:16Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This article investigates the full Boltzmann equation up to second order in the cosmological perturbations. Describing the distribution of polarized radiation by a tensor valued distribution function, we study the gauge dependence of the distribution function and summarize the construction of the gauge-invariant distribution function. The Liouville operator which describes the free streaming of electrons, and the collision term which describes the scattering of photons on free electrons are computed up to second order. Finally, the remaining dependence in the direction of the photon momentum i","authors_text":"Cyril Pitrou","cross_cats":["astro-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2008-09-18T17:01:27Z","title":"The radiative transfer at second order: a full treatment of the Boltzmann equation with polarization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0809.3036","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:504d31f10794d93107faa46e8a6f00fc4f2c1a1a27ab3d82ea80f03bffe30330","target":"record","created_at":"2026-05-18T00:12:16Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"215e9f3033adf6e25d71d113d182f99ba183fcbfdfbe0b3b2c1d9c5196a38d99","cross_cats_sorted":["astro-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2008-09-18T17:01:27Z","title_canon_sha256":"94c5450707254b17bede0a1c18388eaa949ea15ddab40d6e983fea5c79269e49"},"schema_version":"1.0","source":{"id":"0809.3036","kind":"arxiv","version":4}},"canonical_sha256":"6eb6a1679ab93e0c3349a69ced83e6dbb1adb7bd5b0ccf144bf58f7fdc434194","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6eb6a1679ab93e0c3349a69ced83e6dbb1adb7bd5b0ccf144bf58f7fdc434194","first_computed_at":"2026-05-18T00:12:16.388199Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:12:16.388199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mWRwCEGA8dGYtC4+WT4YVstcfp5irDeIKrlC7cIiUDUfuOZuQRa+BSnGcxWzPWMDgeGEXXjbKhjndoUD8TweCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:12:16.388703Z","signed_message":"canonical_sha256_bytes"},"source_id":"0809.3036","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:504d31f10794d93107faa46e8a6f00fc4f2c1a1a27ab3d82ea80f03bffe30330","sha256:a8bb8c3cb42d42aeba4cf28c4f32a360b781db9d1078073e862fe762738cb549"],"state_sha256":"ede689edd92feba22ad08814590161527e1610faa1deab8de7a44a4ccde302a2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T2Ivw9yqDEaSp7Zz5V0vn+uN7XX3vsfhfTQoF0ybt6jJeHBC3mID3IzLachVKupFwCJ7kC+al+LoHD62nf+1Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T08:30:03.438102Z","bundle_sha256":"5939e074b44b6478491dfcd3a5cbf0d67fb58247a03554d934039fe0db737130"}}