Use of the direct translation act of notoriety is in evidence online, and if notaires.fr use it that is a good enough reference in my opinion.
Somebody else asked the question here some years ago and the selected answer was Notarial Certificate, but this is too non-specific IMO, as is notarial act, etc.

NOT FOR GRADING. This "answer" provides substantiation for Yassine's earlier answer.

On appelle « acte de notoriété » un document établi par un notaire qui atteste de votre qualité d'héritier"
https://www.ekie.co/fiches-pratiques/comment-obtenir-un-acte...

"document établi par un notaire qui permet d'identifier tous les héritiers et de préciser leur lien de parenté avec le défunt"
https://www.service-public.fr/particuliers/vosdroits/F12697

"In France, matters of succession are dealt with by notaries. Their involvement is mandatory if the estate includes immovable property. It is optional if there is no immovable property.
The notary draws up the order of succession in a statutory declaration (acte de notoriété)and immovable property certificates recording the transfer of immovable property after the death."
https://e-justice.europa.eu/166/EN/succession?FRANCE&member=...

Although it does not apply to the UK (your case?), the following, which is valid in the EU, might make things 'more easy' for you:
"Proof of Inheritance
In order to be vested, the heir must justify his inheritance. The proof of succession is free and may be established by any means (art. 730 French Civil Code). In practice, it is established by notarial Deed (“acte de notoriété”) referring to the death certificate, the will if any, and the relevant supporting documents (as well as the record of civil status – the family record book). By means of this Deed, a person can easily prove his rights in the estate. The affirmation prevails unless evidence is presented to the contrary (art. 730-3 French Civil Code). Since 17 August 2015 the European Certificate of Succession is used by heirs, legatees having direct rights in the succession and executors of wills or administrators of the estate who, in another Member State, need to invoke their status or to exercise respectively their rights as heirs or legatees and/or their powers as executors of wills or administrators of the estate. This certificate is valid in every Member State of the European Union except Denmark, United Kingdom and Ireland. Now it is more easy to establish the proof of inheritance in Europe."
https://www.ibanet.org/MediaHandler?id=467B235E-8AA9-4BB3-A2...

"The Regulation also introduces a European Certificate of Succession (ECS). The ECS is issued by the authority dealing with the succession, and can be used by heirs, legatees, executors of wills and administrators of the estate to prove their status and exercise their rights or powers in other Member States. Once issued, the ECS will be recognized in all Member States without any special procedure being required."
https://e-justice.europa.eu/478/EN/european_certificate_of_s...

'Succession certificate' works in India too:
"Property owners, who expire without leaving a will, are said to have died intestate. In such cases, the family has to get a succession certificate, which certifies the successor of the deceased. As per sections 380, 381 and 382 of the Indian Succession Act, 1925, a succession certificate holder can take responsibility of the deceased’s assets in addition to any debts he had. The person is then entitled to claim the assets, as per the succession laws. The succession certificate is mandatory, for claiming all kinds of immovable and movable assets, such as bank balance, fixed deposits, investments, etc."
https://housing.com/news/succession-certificate/

I recommend that if you go for 'certificate of succession' you leave the French, just in case there is at any point, in the document itself or in its application, a possibility of confusion with France’s déclaration de succession, a declaration one must file with the tax authorities when receiving an inheritance ( imprimés 2705, 2705-S et 2706): "La succession d'une personne s'ouvre à son décès. Si vous acceptez la succession, le dépôt d'une déclaration est obligatoire." https://www.impots.gouv.fr/formulaire/2705-sd/declaration-de...

There are various ways to translate this as it has no direct equivalent in English/Common law.

It is a notarial certificate which sets out, usually on the sworn statements of persons related to or close to the deceased:
1. Identity of the deceased and where and when he/she died
2.The will (or absence of will)
3. How the deceased's estate devolves
4. Identity of all known heirs and legatees

--------------------------------------------------
Note added at 5 hrs (2024-01-11 18:47:33 GMT)
--------------------------------------------------

VENDRELLA v. ASTRIAB FAMILY LTD. PARTNERSHIP
Connecticut Judicial Branch (.gov)
https://www.jud.ct.gov › supapp › Cases › AROcr
PDF
take judicial notice of matters of common knowledge, the trial court should take judicial notice of the fact that horses have a natural propensity to nip ...
13 pages

HUDOC
https://hudoc.echr.coe.int › fre
... matters of common knowledge, guiding the applicant in providing the relevant information and adequately verifying facts alleged which can be substantiated.

--------------------------------------------------
Note added at 23 hrs (2024-01-12 12:43:18 GMT)
--------------------------------------------------

Explained here:
https://www.capital.fr/votre-argent/acte-de-notoriete-133328...