$\log_{10} \omega_{\text{lin}} = 1.05$ for the linear oscillations, $\mathcal{A}_{\text{log}} = 0.014$ and $\log_{10} \omega_{\text{log}} = 1.26$ for the logarithmic oscillations.

*Acknowledgements.* MBall acknowledges financial support from the INFN InDark initiative and from the COSMOS network (www.cosmosnet.it) through the ASI (Italian Space Agency) Grants 2016-24-H.0 and 2016-24-H.1-2018, as well as 2020-9-HH.0 (participation in LiteBIRD phase A). YA acknowledges support by the Spanish Research Agency (Agencia Estatal de Investigación)'s grant RYC2020-030193-I/AEI/10.13039/501100011033 and the European Social Fund (Fondo Social Europeo) through the Ramón y Cajal program within the State Plan for Scientific and Technical Research and Innovation (Plan Estatal de Investigación Científica y Técnica y de Innovación) 2017-2020, and by the Spanish Research Agency through the grant IFT Centro de Excelencia Severo Ochoa No CEX2020-001007-S funded by MCIN/AEI/10.13039/501100011033. DK was supported by the South African Radio Astronomy Observatory and the National Research Foundation (Grant No. 75415). BL is supported by the European Research Council (ERC) through a starting Grant (ERC-StG-716532 PUNCA), and the UK Science and Technology Facilities Council (STFC) Consolidated Grant No. ST/I00162X/1 and ST/P000541/1. ZS acknowledges ITP institute funding from DFG project 456622116 and IRAP lab support. DS acknowledges financial support from the Fondecyt Regular project number 1200171. MBald acknowledges support by the project 'Combining Cosmic Microwave Background and Large Scale Structure data: an Integrated Approach for Addressing Fundamental Questions in Cosmology', funded by the MIUR Progetti di Ricerca di Rilevate Interesse Nazionale (PRIN) Bando 2017 - grant 2017YJYZAH. GCH acknowledges support through the ESA research fellowship programme. HAW acknowledges support from the Research Council of Norway through project 325113. MV acknowledges partial financial support from the INFN PD51 INDARK grant and from the agreement ASI-INAF n.2017-14-H.0. Part of the N-body simulations presented in this work were obtained on the Ulysses supercomputer at SISSA. JJ acknowledges support by the Swedish Research Council (VR) under the project 2020-05143 - 'Deciphering the Dynamics of Cosmic Structure'. GL acknowledges support by the ANR BIG4 project, grant ANR-16-CE23-0002 of the French Agence Nationale de la Recherche. AA, JJ, and GL acknowledge the support of the Simons Collaboration on "Learning the Universe". DP acknowledges financial support by agreement n. 2020-9-HH.0 ASI-UniRM2. JV was supported by Ruth och Nils-Erik Stenbäcks Stiftelse. The Euclid Consortium acknowledges the European Space Agency and a number of agencies and institutes that have supported the development of Euclid, in particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian Science Policy, the Canadian Euclid Consortium, the French Centre National d'Etudes Spatiales, the Deutsches Zentrum für Luft- und Raumfahrt, the Danish Space Research Institute, the Fundação para a Ciência e a Tecnologia, the Ministerio de Ciencia e Innovación, the National Aeronautics and Space Administration, the National Astronomical Observatory of Japan, the Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Romanian Space Agency, the State Secretariat for Education, Research and Innovation (SERI) at the Swiss Space Office (SSO), and the United Kingdom Space Agency. A complete and detailed list is available on the Euclid web site (http://www.euclid-ec.org).

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$\log_{10} \omega_{\text{lin}} = 1.05$ for the linear oscillations, $\mathcal{A}_{\text{log}} = 0.014$ and $\log_{10} \omega_{\text{log}} = 1.26$ for the logarithmic oscillations.

*Acknowledgements.* MBall acknowledges financial support from the INFN InDark initiative and from the COSMOS network (www.cosmosnet.it) through the ASI (Italian Space Agency) Grants 2016-24-H.0 and 2016-24-H.1-2018, as well as 2020-9-HH.0 (participation in LiteBIRD phase A). YA acknowledges support by the Spanish Research Agency (Agencia Estatal de Investigación)'s grant RYC2020-030193-I/AEI/10.13039/501100011033 and the European Social Fund (Fondo Social Europeo) through the Ramón y Cajal program within the State Plan for Scientific and Technical Research and Innovation (Plan Estatal de Investigación Científica y Técnica y de Innovación) 2017-2020, and by the Spanish Research Agency through the grant IFT Centro de Excelencia Severo Ochoa No CEX2020-001007-S funded by MCIN/AEI/10.13039/501100011033. DK was supported by the South African Radio Astronomy Observatory and the National Research Foundation (Grant No. 75415). BL is supported by the European Research Council (ERC) through a starting Grant (ERC-StG-716532 PUNCA), and the UK Science and Technology Facilities Council (STFC) Consolidated Grant No. ST/I00162X/1 and ST/P000541/1. ZS acknowledges ITP institute funding from DFG project 456622116 and IRAP lab support. DS acknowledges financial support from the Fondecyt Regular project number 1200171. MBald acknowledges support by the project 'Combining Cosmic Microwave Background and Large Scale Structure data: an Integrated Approach for Addressing Fundamental Questions in Cosmology', funded by the MIUR Progetti di Ricerca di Rilevate Interesse Nazionale (PRIN) Bando 2017 - grant 2017YJYZAH. GCH acknowledges support through the ESA research fellowship programme. HAW acknowledges support from the Research Council of Norway through project 325113. MV acknowledges partial financial support from the INFN PD51 INDARK grant and from the agreement ASI-INAF n.2017-14-H.0. Part of the N-body simulations presented in this work were obtained on the Ulysses supercomputer at SISSA. JJ acknowledges support by the Swedish Research Council (VR) under the project 2020-05143 - 'Deciphering the Dynamics of Cosmic Structure'. GL acknowledges support by the ANR BIG4 project, grant ANR-16-CE23-0002 of the French Agence Nationale de la Recherche. AA, JJ, and GL acknowledge the support of the Simons Collaboration on "Learning the Universe". DP acknowledges financial support by agreement n. 2020-9-HH.0 ASI-UniRM2. JV was supported by Ruth och Nils-Erik Stenbäcks Stiftelse. The Euclid Consortium acknowledges the European Space Agency and a number of agencies and institutes that have supported the development of Euclid, in particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian Science Policy, the Canadian Euclid Consortium, the French Centre National d'Etudes Spatiales, the Deutsches Zentrum für Luft- und Raumfahrt, the Danish Space Research Institute, the Fundação para a Ciência e a Tecnologia, the Ministerio de Ciencia e Innovación, the National Aeronautics and Space Administration, the National Astronomical Observatory of Japan, the Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Romanian Space Agency, the State Secretariat for Education, Research and Innovation (SERI) at the Swiss Space Office (SSO), and the United Kingdom Space Agency. A complete and detailed list is available on the Euclid web site (http://www.euclid-ec.org).

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<footnote n="1">¹ Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy e-mail: mario.ballardini@unife.it</footnote>

<footnote n="2">² Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy</footnote>

<footnote n="3">³ INAF – Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy</footnote>

<footnote n="4">⁴ Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain</footnote>

<footnote n="5">⁵ CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA</footnote>

<footnote n="6">⁶ INFN – Bologna, Via Irnerio 46, 40126 Bologna, Italy</footnote>

<footnote n="7">⁷ Department of Physics and Astronomy, University of the Western Cape, Bellville, Cape Town 7535, South Africa</footnote>
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<footnote n="1">¹ Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy e-mail: mario.ballardini@unife.it</footnote>

<footnote n="2">² Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy</footnote>

<footnote n="3">³ INAF – Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy</footnote>

<footnote n="4">⁴ Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain</footnote>

<footnote n="5">⁵ CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA</footnote>

<footnote n="6">⁶ INFN – Bologna, Via Irnerio 46, 40126 Bologna, Italy</footnote>

<footnote n="7">⁷ Department of Physics and Astronomy, University of the Western Cape, Bellville, Cape Town 7535, South Africa</footnote>
⁸ Department of Physics, Institute for Computational Cosmology, Durham University, South Road, DH1 3LE, UK

⁹ Department of Physics & Astronomy, University of Sussex, Brighton BN1 9QH, UK

¹⁰ Université St Joseph; Faculty of Sciences, Beirut, Lebanon

¹¹ Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France

¹² Institut für Theoretische Physik, University of Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

¹³ Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile

¹⁴ Institute Lorentz, Leiden University, PO Box 9506, Leiden 2300 RA, The Netherlands

¹⁵ Departamento de Física, Universidad del País Vasco UPV-EHU, 48940 Leioa, Spain

¹⁶ Dipartimento di Fisica e Astronomia, Università di Bologna, Via Gobetti 93/2, 40129 Bologna, Italy

¹⁷ INFN – Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy

¹⁸ Dipartimento di Fisica e Astronomia “G. Galilei”, Università di Padova, Via Marzolo 8, 35131 Padova, Italy

¹⁹ INFN – Padova, Via Marzolo 8, 35131 Padova, Italy

²⁰ INAF – Osservatorio Astronomico di Padova, Via dell’Osservatorio 5, 35122 Padova, Italy

²¹ European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands

²² Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany

²³ Gran Sasso Science Institute (GSSI), Viale F. Crispi 7, L’Aquila, (AQ) 67100, Italy

²⁴ INFN – Laboratori Nazionali del Gran Sasso (LNGS), L’Aquila, (AQ) 67100, Italy

²⁵ Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029, Blindern 0315, Oslo, Norway

²⁶ IFPU, Institute for Fundamental Physics of the Universe, Via Beirut 2, 34151 Trieste, Italy

²⁷ INAF – Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy

²⁸ SISSA, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, TS, Italy

²⁹ INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, TS, Italy

³⁰ Department of Astronomy, Stockholm University, Albanova 10691 Stockholm, Sweden

³¹ Institut d'Astrophysique de Paris, 98bis Boulevard Arago, 75014 Paris, France

³² Institut d'Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France

³³ The Institute of Mathematical Sciences, HBNI, CIT Campus, Chennai 600113, India

³⁴ Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India

³⁵ Department of Physics, PO Box 64, 00014 University of Helsinki, Finland

³⁶ Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, Finland

³⁷ Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK

³⁸ INAF – Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy

³⁹ INAF – Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese, (TO), Italy

⁴⁰ Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146 Genova, Italy

⁴¹ INFN – Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy

⁴² Department of Physics “E. Pancini”, University Federico II, Via Cinthia 6, 80126 Napoli, Italy

⁴³ INAF – Osservatorio Astronomico di Capodimonte, Via Moariello 16, 80131 Napoli, Italy

⁴⁴ INFN section of Naples, Via Cinthia 6, 80126 Napoli, Italy

⁴⁵ Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal

⁴⁶ Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy

⁴⁷ INFN – Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy

⁴⁸ INAF – IASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy

⁴⁹ Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra, (Barcelona), Spain

⁵⁰ Port d'Informació Científica, Campus UAB, C. Albareda s/n, 08193 Bellaterra, (Barcelona), Spain

⁵¹ INAF – Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy

⁵² Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Università di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy

⁵³ Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK

⁵⁴ European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma, Italy

⁵⁵ ESAC/ESA, Camino Bajo del Castillo s/n., Urb. Villafranca del Castillo, 28692 Villanueva de la Cañada, Madrid, Spain

⁵⁶ University of Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, 69622 Villeurbanne, France

⁵⁷ Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland

⁵⁸ UCB Lyon 1, CNRS/IN2P3, IUF, IP2I Lyon, 4 rue Enrico Fermi, 69622 Villeurbanne, France

⁵⁹ Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande, 1749-016 Lisboa, Portugal

⁶⁰ Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal

⁶¹ Department of Astronomy, University of Geneva, ch. d'Ecogia 16, 1290 Versoix, Switzerland

⁶² INAF – Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere, 100, 00100 Roma, Italy

⁶³ Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191 Gif-sur-Yvette, France

⁶⁴ Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany

⁶⁵ University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany

⁶⁶ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA

⁶⁷ von Hoerner & Sulger GmbH, SchloßPlatz 8, 68723 Schwetzingen, Germany

⁶⁸ Technical University of Denmark, Elektrovej 327, 2800 Kgs., Lyn-gby, Denmark

⁶⁹ Cosmic Dawn Center (DAWN), Copenhagen, Denmark

⁷⁰ Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany

⁷¹ Aix-Marseille Université, CNRS/IN2P3, CPPM, Marseille, France

⁷² Université de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, 1211 Genève 4, Switzerland

⁷³ NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands

⁷⁴ Universität Bonn, Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany

⁷⁵ Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France

⁷⁶ Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Università di Bologna, Via Piero Gobetti 93/2, 40129 Bologna, Italy

⁷⁷ Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris, France

⁷⁸ Department of Physics and Astronomy, University of Aarhus, Ny Munkegade 120, 8000 Aarhus C, Denmark
⁸ Department of Physics, Institute for Computational Cosmology, Durham University, South Road, DH1 3LE, UK

⁹ Department of Physics & Astronomy, University of Sussex, Brighton BN1 9QH, UK

¹⁰ Université St Joseph; Faculty of Sciences, Beirut, Lebanon

¹¹ Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France

¹² Institut für Theoretische Physik, University of Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

¹³ Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile

¹⁴ Institute Lorentz, Leiden University, PO Box 9506, Leiden 2300 RA, The Netherlands

¹⁵ Departamento de Física, Universidad del País Vasco UPV-EHU, 48940 Leioa, Spain

¹⁶ Dipartimento di Fisica e Astronomia, Università di Bologna, Via Gobetti 93/2, 40129 Bologna, Italy

¹⁷ INFN – Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy

¹⁸ Dipartimento di Fisica e Astronomia “G. Galilei”, Università di Padova, Via Marzolo 8, 35131 Padova, Italy

¹⁹ INFN – Padova, Via Marzolo 8, 35131 Padova, Italy

²⁰ INAF – Osservatorio Astronomico di Padova, Via dell’Osservatorio 5, 35122 Padova, Italy

²¹ European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands

²² Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany

²³ Gran Sasso Science Institute (GSSI), Viale F. Crispi 7, L’Aquila, (AQ) 67100, Italy

²⁴ INFN – Laboratori Nazionali del Gran Sasso (LNGS), L’Aquila, (AQ) 67100, Italy

²⁵ Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029, Blindern 0315, Oslo, Norway

²⁶ IFPU, Institute for Fundamental Physics of the Universe, Via Beirut 2, 34151 Trieste, Italy

²⁷ INAF – Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy

²⁸ SISSA, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, TS, Italy

²⁹ INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, TS, Italy

³⁰ Department of Astronomy, Stockholm University, Albanova 10691 Stockholm, Sweden

³¹ Institut d'Astrophysique de Paris, 98bis Boulevard Arago, 75014 Paris, France

³² Institut d'Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France

³³ The Institute of Mathematical Sciences, HBNI, CIT Campus, Chennai 600113, India

³⁴ Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India

³⁵ Department of Physics, PO Box 64, 00014 University of Helsinki, Finland

³⁶ Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, Finland

³⁷ Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK

³⁸ INAF – Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy

³⁹ INAF – Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese, (TO), Italy

⁴⁰ Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146 Genova, Italy

⁴¹ INFN – Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy

⁴² Department of Physics “E. Pancini”, University Federico II, Via Cinthia 6, 80126 Napoli, Italy

⁴³ INAF – Osservatorio Astronomico di Capodimonte, Via Moariello 16, 80131 Napoli, Italy

⁴⁴ INFN section of Naples, Via Cinthia 6, 80126 Napoli, Italy

⁴⁵ Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal

⁴⁶ Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy

⁴⁷ INFN – Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy

⁴⁸ INAF – IASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy

⁴⁹ Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra, (Barcelona), Spain

⁵⁰ Port d'Informació Científica, Campus UAB, C. Albareda s/n, 08193 Bellaterra, (Barcelona), Spain

⁵¹ INAF – Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy

⁵² Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Università di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy

⁵³ Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK

⁵⁴ European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma, Italy

⁵⁵ ESAC/ESA, Camino Bajo del Castillo s/n., Urb. Villafranca del Castillo, 28692 Villanueva de la Cañada, Madrid, Spain

⁵⁶ University of Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, 69622 Villeurbanne, France

⁵⁷ Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland

⁵⁸ UCB Lyon 1, CNRS/IN2P3, IUF, IP2I Lyon, 4 rue Enrico Fermi, 69622 Villeurbanne, France

⁵⁹ Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande, 1749-016 Lisboa, Portugal

⁶⁰ Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal

⁶¹ Department of Astronomy, University of Geneva, ch. d'Ecogia 16, 1290 Versoix, Switzerland

⁶² INAF – Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere, 100, 00100 Roma, Italy

⁶³ Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191 Gif-sur-Yvette, France

⁶⁴ Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany

⁶⁵ University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany

⁶⁶ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA

⁶⁷ von Hoerner & Sulger GmbH, SchloßPlatz 8, 68723 Schwetzingen, Germany

⁶⁸ Technical University of Denmark, Elektrovej 327, 2800 Kgs., Lyn-gby, Denmark

⁶⁹ Cosmic Dawn Center (DAWN), Copenhagen, Denmark

⁷⁰ Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany

⁷¹ Aix-Marseille Université, CNRS/IN2P3, CPPM, Marseille, France

⁷² Université de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, 1211 Genève 4, Switzerland

⁷³ NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands

⁷⁴ Universität Bonn, Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany

⁷⁵ Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France

⁷⁶ Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Università di Bologna, Via Piero Gobetti 93/2, 40129 Bologna, Italy

⁷⁷ Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris, France

⁷⁸ Department of Physics and Astronomy, University of Aarhus, Ny Munkegade 120, 8000 Aarhus C, Denmark
79 Centre for Astrophysics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

80 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

81 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

82 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, Astrophysique, Instrumentation et Modélisation Paris-Saclay, 91191 Gif-sur-Yvette, France

83 Space Science Data Center, Italian Space Agency, Via del Politecnico snc, 00133 Roma, Italy

84 Centre National d'Etudes Spatiales - Centre spatial de Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France

85 Institute of Space Science, Str. Atomistilor, nr. 409 Măgurele, Ilfov 077125, Romania

86 Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstrasse 1, 81679 München, Germany

87 Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr. 25/8, 6020 Innsbruck, Austria

88 Institut d'Estudis Espacials de Catalunya (IEEC), Carrer Gran Capità 2-4, 08034 Barcelona, Spain

89 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain

90 Satlantis, University Science Park, Sede Bld 48940, Leioa-Bilbao, Spain

91 AIM, CEA, CNRS, Université Paris-Saclay, Université de Paris, 91191 Gif-sur-Yvette, France

92 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain

93 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda 1349-018 Lisboa, Portugal

94 Universidad Politécnica de Cartagena, Departamento de Electrónica y Tecnología de Computadoras, Plaza del Hospital 1, 30202 Cartagena, Spain

95 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands

96 Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA

97 Junia, EPA department, 41 Bd Vauban, 59800 Lille, France
79 Centre for Astrophysics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

80 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

81 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

82 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, Astrophysique, Instrumentation et Modélisation Paris-Saclay, 91191 Gif-sur-Yvette, France

83 Space Science Data Center, Italian Space Agency, Via del Politecnico snc, 00133 Roma, Italy

84 Centre National d'Etudes Spatiales - Centre spatial de Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France

85 Institute of Space Science, Str. Atomistilor, nr. 409 Măgurele, Ilfov 077125, Romania

86 Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstrasse 1, 81679 München, Germany

87 Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr. 25/8, 6020 Innsbruck, Austria

88 Institut d'Estudis Espacials de Catalunya (IEEC), Carrer Gran Capità 2-4, 08034 Barcelona, Spain

89 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain

90 Satlantis, University Science Park, Sede Bld 48940, Leioa-Bilbao, Spain

91 AIM, CEA, CNRS, Université Paris-Saclay, Université de Paris, 91191 Gif-sur-Yvette, France

92 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain

93 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda 1349-018 Lisboa, Portugal

94 Universidad Politécnica de Cartagena, Departamento de Electrónica y Tecnología de Computadoras, Plaza del Hospital 1, 30202 Cartagena, Spain

95 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands

96 Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA

97 Junia, EPA department, 41 Bd Vauban, 59800 Lille, France
# Appendix A: Comparison with dark matter N-body simulations

In Figs. A.1 and A.2, we compare the ratio between the matter power spectrum with primordial oscillations to the one obtained

with a power-law PPS at redshift $z = 0$. We show analytic results obtained with PT at LO (blue) and NLO (orange) versus linear theory predictions (grey) and the nonlinear results obtained from N-body simulations (green).

<figure><img src="image_8.png" /><figcaption>**Fig. A.1.** Ratio of IR resummed matter power spectrum at LO (blue) and NLO (orange) obtained for the linear oscillations to the one obtained with a power-law PPS at redshift $z = 0$ varying the IR separation scale $k_S = \epsilon k$ with $\epsilon \in (0.3, 0.7)$. Also shown are the linear results (grey) and the ones obtained from N-body simulations (green).</figcaption></figure>
# Appendix A: Comparison with dark matter N-body simulations

In Figs. A.1 and A.2, we compare the ratio between the matter power spectrum with primordial oscillations to the one obtained

with a power-law PPS at redshift $z = 0$. We show analytic results obtained with PT at LO (blue) and NLO (orange) versus linear theory predictions (grey) and the nonlinear results obtained from N-body simulations (green).

<figure><img src="image_8.png" /><figcaption>**Fig. A.1.** Ratio of IR resummed matter power spectrum at LO (blue) and NLO (orange) obtained for the linear oscillations to the one obtained with a power-law PPS at redshift $z = 0$ varying the IR separation scale $k_S = \epsilon k$ with $\epsilon \in (0.3, 0.7)$. Also shown are the linear results (grey) and the ones obtained from N-body simulations (green).</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. A.2. As in Fig. A.1, but for the logarithmic oscillations.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. A.2. As in Fig. A.1, but for the logarithmic oscillations.</figcaption></figure>
**Appendix B: Nonlinear reconstruction results**

In Fig. B.1, we compare the wiggle $O_w(k)$, which is the ratio between the wiggled and no-wiggle $P(k)$, for all 20 wiggled models: linear models in the first two rows and logarithmic models in the bottom two rows as shown by the legend. The four subpanels of each row, from bottom to top, are the results for $z = 1.5, 1.0, 0.5$, and 0. Linear theory predictions (which in practice are replaced by the measured $P(k)$ from the initial condition snapshots at $z = 9$) are shown by black solid curves, while the measurements from the fully nonlinear and reconstructed mat-

ter density fields are shown by blue solid and orange dashed curves.

Structure formation greatly damps the wiggles, and this impact is strongest for the low-frequency wiggle models, given that there are fewer peaks in the *k* range shown. The extreme example is $\omega_{\text{log}} = 10^{0.2}$, for which at $z < 1$ the only peak in the entire *k* range shown here is completely invisible. Reconstruction is expected to be useful here, as shown by the orange curves in the corresponding panels. Even for models with very high wiggle frequencies, such as $\omega_{\text{log}} = 10^2$, the wiggles can be well reconstructed down to $k \approx 1 h \text{ Mpc}^{-1}$.

<figure><img src="image_10.png" /><figcaption>**Fig. B.1.** Ratio of the linear (black solid curves), nonlinear (blue solid curves), and reconstructed (orange dashed curves) density fields for the 20 selected feature models to the one obtained with a power-law PPS. Each of the rows is respectively for the redshifts marked on the right. The linear $O_w(k)$ are measured from the initial conditions at $z = 9$, the nonlinear $O_w(k)$ are from the output snapshots of DM, and the reconstructed $O_w(k)$ are from the reconstructed density fields.</figcaption></figure>
**Appendix B: Nonlinear reconstruction results**

In Fig. B.1, we compare the wiggle $O_w(k)$, which is the ratio between the wiggled and no-wiggle $P(k)$, for all 20 wiggled models: linear models in the first two rows and logarithmic models in the bottom two rows as shown by the legend. The four subpanels of each row, from bottom to top, are the results for $z = 1.5, 1.0, 0.5$, and 0. Linear theory predictions (which in practice are replaced by the measured $P(k)$ from the initial condition snapshots at $z = 9$) are shown by black solid curves, while the measurements from the fully nonlinear and reconstructed mat-

ter density fields are shown by blue solid and orange dashed curves.

Structure formation greatly damps the wiggles, and this impact is strongest for the low-frequency wiggle models, given that there are fewer peaks in the *k* range shown. The extreme example is $\omega_{\text{log}} = 10^{0.2}$, for which at $z < 1$ the only peak in the entire *k* range shown here is completely invisible. Reconstruction is expected to be useful here, as shown by the orange curves in the corresponding panels. Even for models with very high wiggle frequencies, such as $\omega_{\text{log}} = 10^2$, the wiggles can be well reconstructed down to $k \approx 1 h \, \text{Mpc}^{-1}$.

<figure><img src="image_10.png" /><figcaption>**Fig. B.1.** Ratio of the linear (black solid curves), nonlinear (blue solid curves), and reconstructed (orange dashed curves) density fields for the 20 selected feature models to the one obtained with a power-law PPS. Each of the rows is respectively for the redshifts marked on the right. The linear $O_w(k)$ are measured from the initial conditions at $z = 9$, the nonlinear $O_w(k)$ are from the output snapshots of DM, and the reconstructed $O_w(k)$ are from the reconstructed density fields.</figcaption></figure>