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We consider the problem in the opposite: given $n_s$ as a function of $N$, what is the inflaton potential $V(\\phi)$. We find that for $n_s-1=-2/N$, $V(\\phi)$ is either $\\tanh^2(\\gamma\\phi/2)$ (\"T-model\") or $\\phi^2$ (chaotic inflation) to the leading order in the slow-roll approximation. $\\gamm"},"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":"1504.07692","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"astro-ph.CO","submitted_at":"2015-04-29T01:20:20Z","cross_cats_sorted":["gr-qc"],"title_canon_sha256":"b4c0cb26ad2774f6ceb8b421573992c2770805bd1c0e1cb72a99b4ce7f53e60a","abstract_canon_sha256":"cd818b5eacac3fb2f84faab56ab5ce3b0dda73f2277dbc153f0ee9ccc2a18e7d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:12.853437Z","signature_b64":"tsgq7pbvDDd6RMApXRvK/KxX0+WvPMrOVxZyFZ7HCwgrD87qJ5mbN4MY3ccOaRnSSjrvEujXPBBz7524XdUzAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5016d37a667309bb6a4055e2edd11d45a5846b8e2d7b29670754b8c64e34f131","last_reissued_at":"2026-05-18T01:34:12.852703Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:12.852703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reconstructing the inflaton potential from the spectral index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["gr-qc"],"primary_cat":"astro-ph.CO","authors_text":"Takeshi Chiba","submitted_at":"2015-04-29T01:20:20Z","abstract_excerpt":"Recent cosmological observations are in good agreement with the scalar spectral index $n_s$ with $n_s-1\\sim -2/N$, where $N$ is the number of e-foldings. Quadratic chaotic model, Starobinsky model and Higgs inflation or $\\alpha$-attractors connecting them are typical examples predicting such a relation. We consider the problem in the opposite: given $n_s$ as a function of $N$, what is the inflaton potential $V(\\phi)$. 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