Pith Number

pith:EHZV3VD5

pith:2026:EHZV3VD5XV5XNORFF4A2MH2UEU

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Ringdown and echoes from compact objects: Debye series and Debye quasinormal modes

Mohamed Ould El Hadj, Sam R. Dolan

A Debye series decomposition of waveforms from compact horizonless bodies converges at early times and reconstructs the full signal including prompt response and echoes.

arxiv:2605.15429 v1 · 2026-05-14 · gr-qc · astro-ph.HE

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Record completeness

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Claims

C1strongest claim

We show that the Debye reconstruction matches well with the exact waveform, and that (unlike the standard QNM reconstruction) it converges even at early times, giving an accurate description of all waveform features including the prompt response.

C2weakest assumption

The assumption that the interior of the compact body permits well-defined successive transmissions and that the quasinormal-mode content of each individual Debye term can be extracted and summed independently without significant cross-channel interference (abstract, paragraph describing the Debye decomposition and its application to Schwarzschild-star spacetimes).

C3one line summary

Introduces Debye series and Debye-QNMs to decompose waveforms from Schwarzschild-star models, achieving early-time convergence and organizing ringdown plus echo packets into individual propagation channels.

References

54 extracted · 54 resolved · 26 Pith anchors

[1] dn” stands for down-going. It replaces the more standard “in

[2] Ringdown of thep= 0Debye terms: D-QNM and cut contributions ByinsertingEq.(70a)intoEq.(68), thepurelyexterior contribution is given by ϕ(0) ℓ(t,r) = 1 2πRe [∫ +∞+ic 0+ic e−iω(t−r∗)P (−) ℓ (ω)dω + ∫ +∞

[3] It receives pole contributions from the zeros ofαin ℓ(ω), denoted byω(α) ℓn, which define the interface D-QNMs

[4] Direct reconstruction of the Debye contribution The Debye contribution of orderp≥1is reconstructed from the frequency-domain integral (cf. Eq. (68)) ϕ(p) ℓ(t,r) = 1 2πRe [∫ +∞+ic 0+ic e−iωuD(p) ℓ(ω)dω

[5] This contribution is defined as the discontinuity between the two lips of the cut, as in Eq

Formal links

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Receipt and verification

First computed	2026-05-20T00:00:58.193610Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

21f35dd47dbd7b76ba252f01a61f5425036a9c023fad3767c00efb120154fc9e

Aliases

arxiv: 2605.15429 · arxiv_version: 2605.15429v1 · doi: 10.48550/arxiv.2605.15429 · pith_short_12: EHZV3VD5XV5X · pith_short_16: EHZV3VD5XV5XNORF · pith_short_8: EHZV3VD5

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/EHZV3VD5XV5XNORFF4A2MH2UEU \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 21f35dd47dbd7b76ba252f01a61f5425036a9c023fad3767c00efb120154fc9e

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "fe4f54ac1613fc196e47b2d5b6c7126f1903c7b14bf261b13a97de2e8a6f41c3",
    "cross_cats_sorted": [
      "astro-ph.HE"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "gr-qc",
    "submitted_at": "2026-05-14T21:23:29Z",
    "title_canon_sha256": "b5b234e05d858ec7a23bcb4179a382db2c95d86a3a6190ecd71c0e4afddff860"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15429",
    "kind": "arxiv",
    "version": 1
  }
}