The Pierre Auger Collaboration

# Cosmological implications of photon-flux upper limits at ultra-high energies
in scenarios of Planckian-interacting massive particles for dark matter

P. Abreu  Laboratório de Instrumentação e Física Experimental de Partículas –
LIP and Instituto Superior Técnico – IST, Universidade de Lisboa – UL, Lisboa,
Portugal  M. Aglietta  Osservatorio Astrofisico di Torino (INAF), Torino,
Italy  INFN, Sezione di Torino, Torino, Italy  J.M. Albury  University of
Adelaide, Adelaide, S.A., Australia  I. Allekotte  Centro Atómico Bariloche
and Instituto Balseiro (CNEA-UNCuyo-CONICET), San Carlos de Bariloche,
Argentina  K. Almeida Cheminant  Institute of Nuclear Physics PAN, Krakow,
Poland  A. Almela  Instituto de Tecnologías en Detección y Astropartículas
(CNEA, CONICET, UNSAM), Buenos Aires, Argentina  Universidad Tecnológica
Nacional – Facultad Regional Buenos Aires, Buenos Aires, Argentina  R. Aloisio
Gran Sasso Science Institute, L’Aquila, Italy  INFN Laboratori Nazionali del
Gran Sasso, Assergi (L’Aquila), Italy  J. Alvarez-Muñiz  Instituto Galego de
Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela,
Santiago de Compostela, Spain  R. Alves Batista  IMAPP, Radboud University
Nijmegen, Nijmegen, The Netherlands  J. Ammerman Yebra  Instituto Galego de
Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela,
Santiago de Compostela, Spain  G.A. Anastasi  Osservatorio Astrofisico di
Torino (INAF), Torino, Italy  INFN, Sezione di Torino, Torino, Italy  L.
Anchordoqui  Department of Physics and Astronomy, Lehman College, City
University of New York, Bronx, NY, USA  B. Andrada  Instituto de Tecnologías
en Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina
S. Andringa  Laboratório de Instrumentação e Física Experimental de Partículas
– LIP and Instituto Superior Técnico – IST, Universidade de Lisboa – UL,
Lisboa, Portugal  C. Aramo  INFN, Sezione di Napoli, Napoli, Italy  P.R.
Araújo Ferreira  RWTH Aachen University, III. Physikalisches Institut A,
Aachen, Germany  E. Arnone  Università Torino, Dipartimento di Fisica, Torino,
Italy  INFN, Sezione di Torino, Torino, Italy  J. C. Arteaga Velázquez
Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán, México
H. Asorey  Instituto de Tecnologías en Detección y Astropartículas (CNEA,
CONICET, UNSAM), Buenos Aires, Argentina  P. Assis  Laboratório de
Instrumentação e Física Experimental de Partículas – LIP and Instituto
Superior Técnico – IST, Universidade de Lisboa – UL, Lisboa, Portugal  G.
Avila  Observatorio Pierre Auger and Comisión Nacional de Energía Atómica,
Malargüe, Argentina  E. Avocone  Università dell’Aquila, Dipartimento di
Scienze Fisiche e Chimiche, L’Aquila, Italy  Gran Sasso Science Institute,
L’Aquila, Italy  A.M. Badescu  University Politehnica of Bucharest, Bucharest,
Romania  A. Bakalova  Institute of Physics of the Czech Academy of Sciences,
Prague, Czech Republic  A. Balaceanu  “Horia Hulubei” National Institute for
Physics and Nuclear Engineering, Bucharest-Magurele, Romania  F. Barbato  Gran
Sasso Science Institute, L’Aquila, Italy  INFN Laboratori Nazionali del Gran
Sasso, Assergi (L’Aquila), Italy  J.A. Bellido  University of Adelaide,
Adelaide, S.A., Australia  Universidad Nacional de San Agustin de Arequipa,
Facultad de Ciencias Naturales y Formales, Arequipa, Peru  C. Berat  Univ.
Grenoble Alpes, CNRS, Grenoble Institute of Engineering Univ. Grenoble Alpes,
LPSC-IN2P3, 38000 Grenoble, France  M.E. Bertaina  Università Torino,
Dipartimento di Fisica, Torino, Italy  INFN, Sezione di Torino, Torino, Italy
G. Bhatta  Institute of Nuclear Physics PAN, Krakow, Poland  P.L. Biermann
Max-Planck-Institut für Radioastronomie, Bonn, Germany  V. Binet  Instituto de
Física de Rosario (IFIR) – CONICET/U.N.R. and Facultad de Ciencias Bioquímicas
y Farmacéuticas U.N.R., Rosario, Argentina  K. Bismark  Karlsruhe Institute of
Technology (KIT), Institute for Experimental Particle Physics, Karlsruhe,
Germany  Instituto de Tecnologías en Detección y Astropartículas (CNEA,
CONICET, UNSAM), Buenos Aires, Argentina  T. Bister  RWTH Aachen University,
III. Physikalisches Institut A, Aachen, Germany  J. Biteau  Université Paris-
Saclay, CNRS/IN2P3, IJCLab, Orsay, France  J. Blazek  Institute of Physics of
the Czech Academy of Sciences, Prague, Czech Republic  C. Bleve  Univ.
Grenoble Alpes, CNRS, Grenoble Institute of Engineering Univ. Grenoble Alpes,
LPSC-IN2P3, 38000 Grenoble, France  J. Blümer  Karlsruhe Institute of
Technology (KIT), Institute for Astroparticle Physics, Karlsruhe, Germany  M.
Boháčová  Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic  D. Boncioli  Università dell’Aquila, Dipartimento di Scienze Fisiche
e Chimiche, L’Aquila, Italy  INFN Laboratori Nazionali del Gran Sasso, Assergi
(L’Aquila), Italy  C. Bonifazi  International Center of Advanced Studies and
Instituto de Ciencias Físicas, ECyT-UNSAM and CONICET, Campus Miguelete – San
Martín, Buenos Aires, Argentina  Universidade Federal do Rio de Janeiro,
Instituto de Física, Rio de Janeiro, RJ, Brazil  L. Bonneau Arbeletche
Universidade Estadual de Campinas, IFGW, Campinas, SP, Brazil  N. Borodai
Institute of Nuclear Physics PAN, Krakow, Poland  A.M. Botti  Instituto de
Tecnologías en Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos
Aires, Argentina  J. Brack  Colorado State University, Fort Collins, CO, USA
T. Bretz  RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany  P.G. Brichetto Orchera  Instituto de Tecnologías en Detección y
Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina  F.L. Briechle
RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany  P.
Buchholz  Universität Siegen, Department Physik – Experimentelle
Teilchenphysik, Siegen, Germany  A. Bueno  Universidad de Granada and
C.A.F.P.E., Granada, Spain  S. Buitink  Vrije Universiteit Brussels, Brussels,
Belgium  M. Buscemi  INFN, Sezione di Catania, Catania, Italy  M. Büsken
Karlsruhe Institute of Technology (KIT), Institute for Experimental Particle
Physics, Karlsruhe, Germany  Instituto de Tecnologías en Detección y
Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina  K.S.
Caballero-Mora  Universidad Autónoma de Chiapas, Tuxtla Gutiérrez, Chiapas,
México  L. Caccianiga  Università di Milano, Dipartimento di Fisica, Milano,
Italy  INFN, Sezione di Milano, Milano, Italy  F. Canfora  IMAPP, Radboud
University Nijmegen, Nijmegen, The Netherlands  Nationaal Instituut voor
Kernfysica en Hoge Energie Fysica (NIKHEF), Science Park, Amsterdam, The
Netherlands  I. Caracas  Bergische Universität Wuppertal, Department of
Physics, Wuppertal, Germany  R. Caruso  Università di Catania, Dipartimento di
Fisica e Astronomia “Ettore Majorana“, Catania, Italy  INFN, Sezione di
Catania, Catania, Italy  A. Castellina  Osservatorio Astrofisico di Torino
(INAF), Torino, Italy  INFN, Sezione di Torino, Torino, Italy  F. Catalani
Universidade de São Paulo, Escola de Engenharia de Lorena, Lorena, SP, Brazil
G. Cataldi  INFN, Sezione di Lecce, Lecce, Italy  L. Cazon  Instituto Galego
de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela,
Santiago de Compostela, Spain  M. Cerda  Observatorio Pierre Auger, Malargüe,
Argentina  J.A. Chinellato  Universidade Estadual de Campinas, IFGW, Campinas,
SP, Brazil  J. Chudoba  Institute of Physics of the Czech Academy of Sciences,
Prague, Czech Republic  L. Chytka  Palacky University, RCPTM, Olomouc, Czech
Republic  R.W. Clay  University of Adelaide, Adelaide, S.A., Australia  A.C.
Cobos Cerutti  Instituto de Tecnologías en Detección y Astropartículas (CNEA,
CONICET, UNSAM), and Universidad Tecnológica Nacional – Facultad Regional
Mendoza (CONICET/CNEA), Mendoza, Argentina  R. Colalillo  Università di Napoli
“Federico II”, Dipartimento di Fisica “Ettore Pancini”, Napoli, Italy  INFN,
Sezione di Napoli, Napoli, Italy  A. Coleman  University of Delaware,
Department of Physics and Astronomy, Bartol Research Institute, Newark, DE,
USA  M.R. Coluccia  INFN, Sezione di Lecce, Lecce, Italy  R. Conceição
Laboratório de Instrumentação e Física Experimental de Partículas – LIP and
Instituto Superior Técnico – IST, Universidade de Lisboa – UL, Lisboa,
Portugal  A. Condorelli  Gran Sasso Science Institute, L’Aquila, Italy  INFN
Laboratori Nazionali del Gran Sasso, Assergi (L’Aquila), Italy  G. Consolati
INFN, Sezione di Milano, Milano, Italy  Politecnico di Milano, Dipartimento di
Scienze e Tecnologie Aerospaziali , Milano, Italy  F. Contreras  Observatorio
Pierre Auger and Comisión Nacional de Energía Atómica, Malargüe, Argentina  F.
Convenga  Karlsruhe Institute of Technology (KIT), Institute for Astroparticle
Physics, Karlsruhe, Germany  D. Correia dos Santos  Universidade Federal
Fluminense, EEIMVR, Volta Redonda, RJ, Brazil  C.E. Covault  Case Western
Reserve University, Cleveland, OH, USA  S. Dasso  Instituto de Astronomía y
Física del Espacio (IAFE, CONICET-UBA), Buenos Aires, Argentina  Departamento
de Física and Departamento de Ciencias de la Atmósfera y los Océanos, FCEyN,
Universidad de Buenos Aires and CONICET, Buenos Aires, Argentina  K. Daumiller
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  B.R. Dawson  University of Adelaide, Adelaide, S.A.,
Australia  J.A. Day  University of Adelaide, Adelaide, S.A., Australia  R.M.
de Almeida  Universidade Federal Fluminense, EEIMVR, Volta Redonda, RJ, Brazil
J. de Jesús  Instituto de Tecnologías en Detección y Astropartículas (CNEA,
CONICET, UNSAM), Buenos Aires, Argentina  Karlsruhe Institute of Technology
(KIT), Institute for Astroparticle Physics, Karlsruhe, Germany  S.J. de Jong
IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands  Nationaal
Instituut voor Kernfysica en Hoge Energie Fysica (NIKHEF), Science Park,
Amsterdam, The Netherlands  J.R.T. de Mello Neto  Universidade Federal do Rio
de Janeiro, Instituto de Física, Rio de Janeiro, RJ, Brazil  Universidade
Federal do Rio de Janeiro (UFRJ), Observatório do Valongo, Rio de Janeiro, RJ,
Brazil  I. De Mitri  Gran Sasso Science Institute, L’Aquila, Italy  INFN
Laboratori Nazionali del Gran Sasso, Assergi (L’Aquila), Italy  J. de Oliveira
Instituto Federal de Educação, Ciência e Tecnologia do Rio de Janeiro (IFRJ),
Brazil  D. de Oliveira Franco  Universidade Estadual de Campinas, IFGW,
Campinas, SP, Brazil  F. de Palma  Università del Salento, Dipartimento di
Matematica e Fisica “E. De Giorgi”, Lecce, Italy  INFN, Sezione di Lecce,
Lecce, Italy  V. de Souza  Universidade de São Paulo, Instituto de Física de
São Carlos, São Carlos, SP, Brazil  E. De Vito  Università del Salento,
Dipartimento di Matematica e Fisica “E. De Giorgi”, Lecce, Italy  INFN,
Sezione di Lecce, Lecce, Italy  A. Del Popolo  Università di Catania,
Dipartimento di Fisica e Astronomia “Ettore Majorana“, Catania, Italy  INFN,
Sezione di Catania, Catania, Italy  M. del Río  Observatorio Pierre Auger and
Comisión Nacional de Energía Atómica, Malargüe, Argentina  O. Deligny
CNRS/IN2P3, IJCLab, Université Paris-Saclay, Orsay, France  L. Deval
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  Instituto de Tecnologías en Detección y Astropartículas
(CNEA, CONICET, UNSAM), Buenos Aires, Argentina  A. di Matteo  INFN, Sezione
di Torino, Torino, Italy  M. Dobre  “Horia Hulubei” National Institute for
Physics and Nuclear Engineering, Bucharest-Magurele, Romania  C. Dobrigkeit
Universidade Estadual de Campinas, IFGW, Campinas, SP, Brazil  J.C. D’Olivo
Universidad Nacional Autónoma de México, México, D.F., México  L.M. Domingues
Mendes  Laboratório de Instrumentação e Física Experimental de Partículas –
LIP and Instituto Superior Técnico – IST, Universidade de Lisboa – UL, Lisboa,
Portugal  R.C. dos Anjos  Universidade Federal do Paraná, Setor Palotina,
Palotina, Brazil  M.T. Dova  IFLP, Universidad Nacional de La Plata and
CONICET, La Plata, Argentina  J. Ebr  Institute of Physics of the Czech
Academy of Sciences, Prague, Czech Republic  R. Engel  Karlsruhe Institute of
Technology (KIT), Institute for Experimental Particle Physics, Karlsruhe,
Germany  Karlsruhe Institute of Technology (KIT), Institute for Astroparticle
Physics, Karlsruhe, Germany  I. Epicoco  Università del Salento, Dipartimento
di Matematica e Fisica “E. De Giorgi”, Lecce, Italy  INFN, Sezione di Lecce,
Lecce, Italy  M. Erdmann  RWTH Aachen University, III. Physikalisches Institut
A, Aachen, Germany  C.O. Escobar  Fermi National Accelerator Laboratory,
Fermilab, Batavia, IL, USA  A. Etchegoyen  Instituto de Tecnologías en
Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina
Universidad Tecnológica Nacional – Facultad Regional Buenos Aires, Buenos
Aires, Argentina  H. Falcke  IMAPP, Radboud University Nijmegen, Nijmegen, The
Netherlands  Stichting Astronomisch Onderzoek in Nederland (ASTRON),
Dwingeloo, The Netherlands  Nationaal Instituut voor Kernfysica en Hoge
Energie Fysica (NIKHEF), Science Park, Amsterdam, The Netherlands  J. Farmer
University of Chicago, Enrico Fermi Institute, Chicago, IL, USA  G. Farrar
New York University, New York, NY, USA  A.C. Fauth  Universidade Estadual de
Campinas, IFGW, Campinas, SP, Brazil  N. Fazzini  Fermi National Accelerator
Laboratory, Fermilab, Batavia, IL, USA  F. Feldbusch  Karlsruhe Institute of
Technology (KIT), Institut für Prozessdatenverarbeitung und Elektronik,
Karlsruhe, Germany  F. Fenu  Università Torino, Dipartimento di Fisica,
Torino, Italy  INFN, Sezione di Torino, Torino, Italy  B. Fick  Michigan
Technological University, Houghton, MI, USA  J.M. Figueira  Instituto de
Tecnologías en Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos
Aires, Argentina  A. Filipčič  Experimental Particle Physics Department, J.
Stefan Institute, Ljubljana, Slovenia  Center for Astrophysics and Cosmology
(CAC), University of Nova Gorica, Nova Gorica, Slovenia  T. Fitoussi
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  T. Fodran  IMAPP, Radboud University Nijmegen, Nijmegen,
The Netherlands  T. Fujii  University of Chicago, Enrico Fermi Institute,
Chicago, IL, USA  now at Hakubi Center for Advanced Research and Graduate
School of Science, Kyoto University, Kyoto, Japan  A. Fuster  Instituto de
Tecnologías en Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos
Aires, Argentina  Universidad Tecnológica Nacional – Facultad Regional Buenos
Aires, Buenos Aires, Argentina  C. Galea  IMAPP, Radboud University Nijmegen,
Nijmegen, The Netherlands  C. Galelli  Università di Milano, Dipartimento di
Fisica, Milano, Italy  INFN, Sezione di Milano, Milano, Italy  B. García
Instituto de Tecnologías en Detección y Astropartículas (CNEA, CONICET,
UNSAM), and Universidad Tecnológica Nacional – Facultad Regional Mendoza
(CONICET/CNEA), Mendoza, Argentina  A.L. Garcia Vegas  RWTH Aachen University,
III. Physikalisches Institut A, Aachen, Germany  H. Gemmeke  Karlsruhe
Institute of Technology (KIT), Institut für Prozessdatenverarbeitung und
Elektronik, Karlsruhe, Germany  F. Gesualdi  Instituto de Tecnologías en
Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  A. Gherghel-Lascu  “Horia Hulubei” National Institute for
Physics and Nuclear Engineering, Bucharest-Magurele, Romania  P.L. Ghia
CNRS/IN2P3, IJCLab, Université Paris-Saclay, Orsay, France  U. Giaccari
IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands  M. Giammarchi
INFN, Sezione di Milano, Milano, Italy  J. Glombitza  RWTH Aachen University,
III. Physikalisches Institut A, Aachen, Germany  F. Gobbi  Observatorio Pierre
Auger, Malargüe, Argentina  F. Gollan  Instituto de Tecnologías en Detección y
Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina  G. Golup
Centro Atómico Bariloche and Instituto Balseiro (CNEA-UNCuyo-CONICET), San
Carlos de Bariloche, Argentina  M. Gómez Berisso  Centro Atómico Bariloche and
Instituto Balseiro (CNEA-UNCuyo-CONICET), San Carlos de Bariloche, Argentina
P.F. Gómez Vitale  Observatorio Pierre Auger and Comisión Nacional de Energía
Atómica, Malargüe, Argentina  J.P. Gongora  Observatorio Pierre Auger and
Comisión Nacional de Energía Atómica, Malargüe, Argentina  J.M. González
Centro Atómico Bariloche and Instituto Balseiro (CNEA-UNCuyo-CONICET), San
Carlos de Bariloche, Argentina  N. González  Université Libre de Bruxelles
(ULB), Brussels, Belgium  I. Goos  Centro Atómico Bariloche and Instituto
Balseiro (CNEA-UNCuyo-CONICET), San Carlos de Bariloche, Argentina  Karlsruhe
Institute of Technology (KIT), Institute for Astroparticle Physics, Karlsruhe,
Germany  D. Góra  Institute of Nuclear Physics PAN, Krakow, Poland  A. Gorgi
Osservatorio Astrofisico di Torino (INAF), Torino, Italy  INFN, Sezione di
Torino, Torino, Italy  M. Gottowik  Bergische Universität Wuppertal,
Department of Physics, Wuppertal, Germany  T.D. Grubb  University of Adelaide,
Adelaide, S.A., Australia  F. Guarino  Università di Napoli “Federico II”,
Dipartimento di Fisica “Ettore Pancini”, Napoli, Italy  INFN, Sezione di
Napoli, Napoli, Italy  G.P. Guedes  Universidade Estadual de Feira de Santana,
Feira de Santana, Brazil  E. Guido  INFN, Sezione di Torino, Torino, Italy
Università Torino, Dipartimento di Fisica, Torino, Italy  S. Hahn  Karlsruhe
Institute of Technology (KIT), Institute for Astroparticle Physics, Karlsruhe,
Germany  Instituto de Tecnologías en Detección y Astropartículas (CNEA,
CONICET, UNSAM), Buenos Aires, Argentina  P. Hamal  Institute of Physics of
the Czech Academy of Sciences, Prague, Czech Republic  M.R. Hampel  Instituto
de Tecnologías en Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos
Aires, Argentina  P. Hansen  IFLP, Universidad Nacional de La Plata and
CONICET, La Plata, Argentina  D. Harari  Centro Atómico Bariloche and
Instituto Balseiro (CNEA-UNCuyo-CONICET), San Carlos de Bariloche, Argentina
V.M. Harvey  University of Adelaide, Adelaide, S.A., Australia  A. Haungs
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  T. Hebbeker  RWTH Aachen University, III. Physikalisches
Institut A, Aachen, Germany  D. Heck  Karlsruhe Institute of Technology (KIT),
Institute for Astroparticle Physics, Karlsruhe, Germany  G.C. Hill  University
of Adelaide, Adelaide, S.A., Australia  C. Hojvat  Fermi National Accelerator
Laboratory, Fermilab, Batavia, IL, USA  J.R. Hörandel  IMAPP, Radboud
University Nijmegen, Nijmegen, The Netherlands  Nationaal Instituut voor
Kernfysica en Hoge Energie Fysica (NIKHEF), Science Park, Amsterdam, The
Netherlands  P. Horvath  Palacky University, RCPTM, Olomouc, Czech Republic
M. Hrabovský  Palacky University, RCPTM, Olomouc, Czech Republic  T. Huege
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  Vrije Universiteit Brussels, Brussels, Belgium  A. Insolia
Università di Catania, Dipartimento di Fisica e Astronomia “Ettore Majorana“,
Catania, Italy  INFN, Sezione di Catania, Catania, Italy  P.G. Isar  Institute
of Space Science, Bucharest-Magurele, Romania  P. Janecek  Institute of
Physics of the Czech Academy of Sciences, Prague, Czech Republic  J.A. Johnsen
Colorado School of Mines, Golden, CO, USA  J. Jurysek  Institute of Physics of
the Czech Academy of Sciences, Prague, Czech Republic  A. Kääpä  Bergische
Universität Wuppertal, Department of Physics, Wuppertal, Germany  K.H. Kampert
Bergische Universität Wuppertal, Department of Physics, Wuppertal, Germany  B.
Keilhauer  Karlsruhe Institute of Technology (KIT), Institute for
Astroparticle Physics, Karlsruhe, Germany  A. Khakurdikar  IMAPP, Radboud
University Nijmegen, Nijmegen, The Netherlands  V.V. Kizakke Covilakam
Instituto de Tecnologías en Detección y Astropartículas (CNEA, CONICET,
UNSAM), Buenos Aires, Argentina  Karlsruhe Institute of Technology (KIT),
Institute for Astroparticle Physics, Karlsruhe, Germany  H.O. Klages
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  M. Kleifges  Karlsruhe Institute of Technology (KIT),
Institut für Prozessdatenverarbeitung und Elektronik, Karlsruhe, Germany  J.
Kleinfeller  Observatorio Pierre Auger, Malargüe, Argentina  F. Knapp
Karlsruhe Institute of Technology (KIT), Institute for Experimental Particle
Physics, Karlsruhe, Germany  N. Kunka  Karlsruhe Institute of Technology
(KIT), Institut für Prozessdatenverarbeitung und Elektronik, Karlsruhe,
Germany  B.L. Lago  Centro Federal de Educação Tecnológica Celso Suckow da
Fonseca, Nova Friburgo, Brazil  N. Langner  RWTH Aachen University, III.
Physikalisches Institut A, Aachen, Germany  M.A. Leigui de Oliveira
Universidade Federal do ABC, Santo André, SP, Brazil  V. Lenok  Karlsruhe
Institute of Technology (KIT), Institute for Astroparticle Physics, Karlsruhe,
Germany  A. Letessier-Selvon  Laboratoire de Physique Nucléaire et de Hautes
Energies (LPNHE), Sorbonne Université, Université de Paris, CNRS-IN2P3, Paris,
France  I. Lhenry-Yvon  CNRS/IN2P3, IJCLab, Université Paris-Saclay, Orsay,
France  D. Lo Presti  Università di Catania, Dipartimento di Fisica e
Astronomia “Ettore Majorana“, Catania, Italy  INFN, Sezione di Catania,
Catania, Italy  L. Lopes  Laboratório de Instrumentação e Física Experimental
de Partículas – LIP and Instituto Superior Técnico – IST, Universidade de
Lisboa – UL, Lisboa, Portugal  R. López  Benemérita Universidad Autónoma de
Puebla, Puebla, México  L. Lu  University of Wisconsin-Madison, Department of
Physics and WIPAC, Madison, WI, USA  Q. Luce  Karlsruhe Institute of
Technology (KIT), Institute for Experimental Particle Physics, Karlsruhe,
Germany  J.P. Lundquist  Center for Astrophysics and Cosmology (CAC),
University of Nova Gorica, Nova Gorica, Slovenia  A. Machado Payeras
Universidade Estadual de Campinas, IFGW, Campinas, SP, Brazil  G. Mancarella
Università del Salento, Dipartimento di Matematica e Fisica “E. De Giorgi”,
Lecce, Italy  INFN, Sezione di Lecce, Lecce, Italy  D. Mandat  Institute of
Physics of the Czech Academy of Sciences, Prague, Czech Republic  B.C. Manning
University of Adelaide, Adelaide, S.A., Australia  J. Manshanden  Universität
Hamburg, II. Institut für Theoretische Physik, Hamburg, Germany  P. Mantsch
Fermi National Accelerator Laboratory, Fermilab, Batavia, IL, USA  S. Marafico
CNRS/IN2P3, IJCLab, Université Paris-Saclay, Orsay, France  F.M. Mariani
Università di Milano, Dipartimento di Fisica, Milano, Italy  INFN, Sezione di
Milano, Milano, Italy  A.G. Mariazzi  IFLP, Universidad Nacional de La Plata
and CONICET, La Plata, Argentina  I.C. Mariş  Université Libre de Bruxelles
(ULB), Brussels, Belgium  G. Marsella  Università di Palermo, Dipartimento di
Fisica e Chimica ”E. Segrè”, Palermo, Italy  INFN, Sezione di Catania,
Catania, Italy  D. Martello  Università del Salento, Dipartimento di
Matematica e Fisica “E. De Giorgi”, Lecce, Italy  INFN, Sezione di Lecce,
Lecce, Italy  S. Martinelli  Karlsruhe Institute of Technology (KIT),
Institute for Astroparticle Physics, Karlsruhe, Germany  Instituto de
Tecnologías en Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos
Aires, Argentina  O. Martínez Bravo  Benemérita Universidad Autónoma de
Puebla, Puebla, México  M. Mastrodicasa  Università dell’Aquila, Dipartimento
di Scienze Fisiche e Chimiche, L’Aquila, Italy  INFN Laboratori Nazionali del
Gran Sasso, Assergi (L’Aquila), Italy  H.J. Mathes  Karlsruhe Institute of
Technology (KIT), Institute for Astroparticle Physics, Karlsruhe, Germany  J.
Matthews  Louisiana State University, Baton Rouge, LA, USA  G. Matthiae
Università di Roma “Tor Vergata”, Dipartimento di Fisica, Roma, Italy  INFN,
Sezione di Roma “Tor Vergata”, Roma, Italy  E. Mayotte  Colorado School of
Mines, Golden, CO, USA  Bergische Universität Wuppertal, Department of
Physics, Wuppertal, Germany  S. Mayotte  Colorado School of Mines, Golden, CO,
USA  P.O. Mazur  Fermi National Accelerator Laboratory, Fermilab, Batavia, IL,
USA  G. Medina-Tanco  Universidad Nacional Autónoma de México, México, D.F.,
México  D. Melo  Instituto de Tecnologías en Detección y Astropartículas
(CNEA, CONICET, UNSAM), Buenos Aires, Argentina  A. Menshikov  Karlsruhe
Institute of Technology (KIT), Institut für Prozessdatenverarbeitung und
Elektronik, Karlsruhe, Germany  S. Michal  Palacky University, RCPTM, Olomouc,
Czech Republic  M.I. Micheletti  Instituto de Física de Rosario (IFIR) –
CONICET/U.N.R. and Facultad de Ciencias Bioquímicas y Farmacéuticas U.N.R.,
Rosario, Argentina  L. Miramonti  Università di Milano, Dipartimento di
Fisica, Milano, Italy  INFN, Sezione di Milano, Milano, Italy  S. Mollerach
Centro Atómico Bariloche and Instituto Balseiro (CNEA-UNCuyo-CONICET), San
Carlos de Bariloche, Argentina  F. Montanet  Univ. Grenoble Alpes, CNRS,
Grenoble Institute of Engineering Univ. Grenoble Alpes, LPSC-IN2P3, 38000
Grenoble, France  L. Morejon  Bergische Universität Wuppertal, Department of
Physics, Wuppertal, Germany  C. Morello  Osservatorio Astrofisico di Torino
(INAF), Torino, Italy  INFN, Sezione di Torino, Torino, Italy  M. Mostafá
Pennsylvania State University, University Park, PA, USA  A.L. Müller
Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic
M.A. Muller  Universidade Estadual de Campinas, IFGW, Campinas, SP, Brazil  K.
Mulrey  IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands
Nationaal Instituut voor Kernfysica en Hoge Energie Fysica (NIKHEF), Science
Park, Amsterdam, The Netherlands  R. Mussa  INFN, Sezione di Torino, Torino,
Italy  M. Muzio  New York University, New York, NY, USA  W.M. Namasaka
Bergische Universität Wuppertal, Department of Physics, Wuppertal, Germany  A.
Nasr-Esfahani  Bergische Universität Wuppertal, Department of Physics,
Wuppertal, Germany  L. Nellen  Universidad Nacional Autónoma de México,
México, D.F., México  G. Nicora  Centro de Investigaciones en Láseres y
Aplicaciones, CITEDEF and CONICET, Villa Martelli, Argentina  M. Niculescu-
Oglinzanu  “Horia Hulubei” National Institute for Physics and Nuclear
Engineering, Bucharest-Magurele, Romania  M. Niechciol  Universität Siegen,
Department Physik – Experimentelle Teilchenphysik, Siegen, Germany  D. Nitz
Michigan Technological University, Houghton, MI, USA  I. Norwood  Michigan
Technological University, Houghton, MI, USA  D. Nosek  Charles University,
Faculty of Mathematics and Physics, Institute of Particle and Nuclear Physics,
Prague, Czech Republic  V. Novotny  Charles University, Faculty of Mathematics
and Physics, Institute of Particle and Nuclear Physics, Prague, Czech Republic
L. Nožka  Palacky University, RCPTM, Olomouc, Czech Republic  A Nucita
Università del Salento, Dipartimento di Matematica e Fisica “E. De Giorgi”,
Lecce, Italy  INFN, Sezione di Lecce, Lecce, Italy  L.A. Núñez  Universidad
Industrial de Santander, Bucaramanga, Colombia  C. Oliveira  Universidade de
São Paulo, Instituto de Física de São Carlos, São Carlos, SP, Brazil  M.
Palatka  Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic  J. Pallotta  Centro de Investigaciones en Láseres y Aplicaciones,
CITEDEF and CONICET, Villa Martelli, Argentina  P. Papenbreer  Bergische
Universität Wuppertal, Department of Physics, Wuppertal, Germany  G. Parente
Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago
de Compostela, Santiago de Compostela, Spain  A. Parra  Benemérita Universidad
Autónoma de Puebla, Puebla, México  J. Pawlowsky  Bergische Universität
Wuppertal, Department of Physics, Wuppertal, Germany  M. Pech  Institute of
Physics of the Czech Academy of Sciences, Prague, Czech Republic  J. Pȩkala
Institute of Nuclear Physics PAN, Krakow, Poland  R. Pelayo  Unidad
Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas del
Instituto Politécnico Nacional (UPIITA-IPN), México, D.F., México  J. Peña-
Rodriguez  Universidad Industrial de Santander, Bucaramanga, Colombia  E.E.
Pereira Martins  Karlsruhe Institute of Technology (KIT), Institute for
Experimental Particle Physics, Karlsruhe, Germany  Instituto de Tecnologías en
Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina
J. Perez Armand  Universidade de São Paulo, Instituto de Física, São Paulo,
SP, Brazil  C. Pérez Bertolli  Instituto de Tecnologías en Detección y
Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina  Karlsruhe
Institute of Technology (KIT), Institute for Astroparticle Physics, Karlsruhe,
Germany  L. Perrone  Università del Salento, Dipartimento di Matematica e
Fisica “E. De Giorgi”, Lecce, Italy  INFN, Sezione di Lecce, Lecce, Italy  S.
Petrera  Gran Sasso Science Institute, L’Aquila, Italy  INFN Laboratori
Nazionali del Gran Sasso, Assergi (L’Aquila), Italy  C. Petrucci  Università
dell’Aquila, Dipartimento di Scienze Fisiche e Chimiche, L’Aquila, Italy  INFN
Laboratori Nazionali del Gran Sasso, Assergi (L’Aquila), Italy  T. Pierog
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  M. Pimenta  Laboratório de Instrumentação e Física
Experimental de Partículas – LIP and Instituto Superior Técnico – IST,
Universidade de Lisboa – UL, Lisboa, Portugal  V. Pirronello  Università di
Catania, Dipartimento di Fisica e Astronomia “Ettore Majorana“, Catania, Italy
INFN, Sezione di Catania, Catania, Italy  M. Platino  Instituto de Tecnologías
en Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina
B. Pont  IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands  M.
Pothast  Nationaal Instituut voor Kernfysica en Hoge Energie Fysica (NIKHEF),
Science Park, Amsterdam, The Netherlands  IMAPP, Radboud University Nijmegen,
Nijmegen, The Netherlands  P. Privitera  University of Chicago, Enrico Fermi
Institute, Chicago, IL, USA  M. Prouza  Institute of Physics of the Czech
Academy of Sciences, Prague, Czech Republic  A. Puyleart  Michigan
Technological University, Houghton, MI, USA  S. Querchfeld  Bergische
Universität Wuppertal, Department of Physics, Wuppertal, Germany  J.
Rautenberg  Bergische Universität Wuppertal, Department of Physics, Wuppertal,
Germany  D. Ravignani  Instituto de Tecnologías en Detección y Astropartículas
(CNEA, CONICET, UNSAM), Buenos Aires, Argentina  M. Reininghaus  Karlsruhe
Institute of Technology (KIT), Institute for Astroparticle Physics, Karlsruhe,
Germany  Instituto de Tecnologías en Detección y Astropartículas (CNEA,
CONICET, UNSAM), Buenos Aires, Argentina  J. Ridky  Institute of Physics of
the Czech Academy of Sciences, Prague, Czech Republic  F. Riehn  Laboratório
de Instrumentação e Física Experimental de Partículas – LIP and Instituto
Superior Técnico – IST, Universidade de Lisboa – UL, Lisboa, Portugal  M.
Risse  Universität Siegen, Department Physik – Experimentelle Teilchenphysik,
Siegen, Germany  V. Rizi  Università dell’Aquila, Dipartimento di Scienze
Fisiche e Chimiche, L’Aquila, Italy  INFN Laboratori Nazionali del Gran Sasso,
Assergi (L’Aquila), Italy  W. Rodrigues de Carvalho  IMAPP, Radboud University
Nijmegen, Nijmegen, The Netherlands  J. Rodriguez Rojo  Observatorio Pierre
Auger and Comisión Nacional de Energía Atómica, Malargüe, Argentina  M.J.
Roncoroni  Instituto de Tecnologías en Detección y Astropartículas (CNEA,
CONICET, UNSAM), Buenos Aires, Argentina  S. Rossoni  Universität Hamburg, II.
Institut für Theoretische Physik, Hamburg, Germany  M. Roth  Karlsruhe
Institute of Technology (KIT), Institute for Astroparticle Physics, Karlsruhe,
Germany  E. Roulet  Centro Atómico Bariloche and Instituto Balseiro (CNEA-
UNCuyo-CONICET), San Carlos de Bariloche, Argentina  A.C. Rovero  Instituto de
Astronomía y Física del Espacio (IAFE, CONICET-UBA), Buenos Aires, Argentina
P. Ruehl  Universität Siegen, Department Physik – Experimentelle
Teilchenphysik, Siegen, Germany  A. Saftoiu  “Horia Hulubei” National
Institute for Physics and Nuclear Engineering, Bucharest-Magurele, Romania  M.
Saharan  IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands  F.
Salamida  Università dell’Aquila, Dipartimento di Scienze Fisiche e Chimiche,
L’Aquila, Italy  INFN Laboratori Nazionali del Gran Sasso, Assergi (L’Aquila),
Italy  H. Salazar  Benemérita Universidad Autónoma de Puebla, Puebla, México
G. Salina  INFN, Sezione di Roma “Tor Vergata”, Roma, Italy  J.D. Sanabria
Gomez  Universidad Industrial de Santander, Bucaramanga, Colombia  F. Sánchez
Instituto de Tecnologías en Detección y Astropartículas (CNEA, CONICET,
UNSAM), Buenos Aires, Argentina  E.M. Santos  Universidade de São Paulo,
Instituto de Física, São Paulo, SP, Brazil  E. Santos  Institute of Physics of
the Czech Academy of Sciences, Prague, Czech Republic  F. Sarazin  Colorado
School of Mines, Golden, CO, USA  R. Sarmento  Laboratório de Instrumentação e
Física Experimental de Partículas – LIP and Instituto Superior Técnico – IST,
Universidade de Lisboa – UL, Lisboa, Portugal  C. Sarmiento-Cano  Instituto de
Tecnologías en Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos
Aires, Argentina  R. Sato  Observatorio Pierre Auger and Comisión Nacional de
Energía Atómica, Malargüe, Argentina  P. Savina  University of Wisconsin-
Madison, Department of Physics and WIPAC, Madison, WI, USA  C.M. Schäfer
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  V. Scherini  Università del Salento, Dipartimento di
Matematica e Fisica “E. De Giorgi”, Lecce, Italy  INFN, Sezione di Lecce,
Lecce, Italy  H. Schieler  Karlsruhe Institute of Technology (KIT), Institute
for Astroparticle Physics, Karlsruhe, Germany  M. Schimassek  Karlsruhe
Institute of Technology (KIT), Institute for Experimental Particle Physics,
Karlsruhe, Germany  Instituto de Tecnologías en Detección y Astropartículas
(CNEA, CONICET, UNSAM), Buenos Aires, Argentina  M. Schimp  Bergische
Universität Wuppertal, Department of Physics, Wuppertal, Germany  F. Schlüter
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  Instituto de Tecnologías en Detección y Astropartículas
(CNEA, CONICET, UNSAM), Buenos Aires, Argentina  D. Schmidt  Karlsruhe
Institute of Technology (KIT), Institute for Experimental Particle Physics,
Karlsruhe, Germany  O. Scholten  Vrije Universiteit Brussels, Brussels,
Belgium  H. Schoorlemmer  IMAPP, Radboud University Nijmegen, Nijmegen, The
Netherlands  Nationaal Instituut voor Kernfysica en Hoge Energie Fysica
(NIKHEF), Science Park, Amsterdam, The Netherlands  P. Schovánek  Institute of
Physics of the Czech Academy of Sciences, Prague, Czech Republic  F.G.
Schröder  University of Delaware, Department of Physics and Astronomy, Bartol
Research Institute, Newark, DE, USA  Karlsruhe Institute of Technology (KIT),
Institute for Astroparticle Physics, Karlsruhe, Germany  J. Schulte  RWTH
Aachen University, III. Physikalisches Institut A, Aachen, Germany  T. Schulz
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  S.J. Sciutto  IFLP, Universidad Nacional de La Plata and
CONICET, La Plata, Argentina  M. Scornavacche  Instituto de Tecnologías en
Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  A. Segreto  Istituto di Astrofisica Spaziale e Fisica
Cosmica di Palermo (INAF), Palermo, Italy  INFN, Sezione di Catania, Catania,
Italy  S. Sehgal  Bergische Universität Wuppertal, Department of Physics,
Wuppertal, Germany  R.C. Shellard  Centro Brasileiro de Pesquisas Fisicas, Rio
de Janeiro, RJ, Brazil  G. Sigl  Universität Hamburg, II. Institut für
Theoretische Physik, Hamburg, Germany  G. Silli  Instituto de Tecnologías en
Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  O. Sima  “Horia Hulubei” National Institute for Physics
and Nuclear Engineering, Bucharest-Magurele, Romania  also at University of
Bucharest, Physics Department, Bucharest, Romania  R. Smau  “Horia Hulubei”
National Institute for Physics and Nuclear Engineering, Bucharest-Magurele,
Romania  R. Šmída  University of Chicago, Enrico Fermi Institute, Chicago, IL,
USA  P. Sommers  Pennsylvania State University, University Park, PA, USA  J.F.
Soriano  Department of Physics and Astronomy, Lehman College, City University
of New York, Bronx, NY, USA  R. Squartini  Observatorio Pierre Auger,
Malargüe, Argentina  M. Stadelmaier  Karlsruhe Institute of Technology (KIT),
Institute for Astroparticle Physics, Karlsruhe, Germany  Instituto de
Tecnologías en Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos
Aires, Argentina  D. Stanca  “Horia Hulubei” National Institute for Physics
and Nuclear Engineering, Bucharest-Magurele, Romania  S. Stanič  Center for
Astrophysics and Cosmology (CAC), University of Nova Gorica, Nova Gorica,
Slovenia  J. Stasielak  Institute of Nuclear Physics PAN, Krakow, Poland  P.
Stassi  Univ. Grenoble Alpes, CNRS, Grenoble Institute of Engineering Univ.
Grenoble Alpes, LPSC-IN2P3, 38000 Grenoble, France  A. Streich  Karlsruhe
Institute of Technology (KIT), Institute for Experimental Particle Physics,
Karlsruhe, Germany  Instituto de Tecnologías en Detección y Astropartículas
(CNEA, CONICET, UNSAM), Buenos Aires, Argentina  M. Suárez-Durán  Université
Libre de Bruxelles (ULB), Brussels, Belgium  T. Sudholz  University of
Adelaide, Adelaide, S.A., Australia  T. Suomijärvi  Université Paris-Saclay,
CNRS/IN2P3, IJCLab, Orsay, France  A.D. Supanitsky  Instituto de Tecnologías
en Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina
Z. Szadkowski  University of Łódź, Faculty of High-Energy Astrophysics,Łódź,
Poland  A. Tapia  Universidad de Medellín, Medellín, Colombia  C. Taricco
Università Torino, Dipartimento di Fisica, Torino, Italy  INFN, Sezione di
Torino, Torino, Italy  C. Timmermans  Nationaal Instituut voor Kernfysica en
Hoge Energie Fysica (NIKHEF), Science Park, Amsterdam, The Netherlands  IMAPP,
Radboud University Nijmegen, Nijmegen, The Netherlands  O. Tkachenko
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  P. Tobiska  Institute of Physics of the Czech Academy of
Sciences, Prague, Czech Republic  C.J. Todero Peixoto  Universidade de São
Paulo, Escola de Engenharia de Lorena, Lorena, SP, Brazil  B. Tomé
Laboratório de Instrumentação e Física Experimental de Partículas – LIP and
Instituto Superior Técnico – IST, Universidade de Lisboa – UL, Lisboa,
Portugal  Z. Torrès  Univ. Grenoble Alpes, CNRS, Grenoble Institute of
Engineering Univ. Grenoble Alpes, LPSC-IN2P3, 38000 Grenoble, France  A.
Travaini  Observatorio Pierre Auger, Malargüe, Argentina  P. Travnicek
Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic
C. Trimarelli  Università dell’Aquila, Dipartimento di Scienze Fisiche e
Chimiche, L’Aquila, Italy  INFN Laboratori Nazionali del Gran Sasso, Assergi
(L’Aquila), Italy  M. Tueros  IFLP, Universidad Nacional de La Plata and
CONICET, La Plata, Argentina  R. Ulrich  Karlsruhe Institute of Technology
(KIT), Institute for Astroparticle Physics, Karlsruhe, Germany  M. Unger
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  L. Vaclavek  Palacky University, RCPTM, Olomouc, Czech
Republic  M. Vacula  Palacky University, RCPTM, Olomouc, Czech Republic  J.F.
Valdés Galicia  Universidad Nacional Autónoma de México, México, D.F., México
L. Valore  Università di Napoli “Federico II”, Dipartimento di Fisica “Ettore
Pancini”, Napoli, Italy  INFN, Sezione di Napoli, Napoli, Italy  E. Varela
Benemérita Universidad Autónoma de Puebla, Puebla, México  A. Vásquez-Ramírez
Universidad Industrial de Santander, Bucaramanga, Colombia  D. Veberič
Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics,
Karlsruhe, Germany  C. Ventura  Universidade Federal do Rio de Janeiro (UFRJ),
Observatório do Valongo, Rio de Janeiro, RJ, Brazil  I.D. Vergara Quispe
IFLP, Universidad Nacional de La Plata and CONICET, La Plata, Argentina  V.
Verzi  INFN, Sezione di Roma “Tor Vergata”, Roma, Italy  J. Vicha  Institute
of Physics of the Czech Academy of Sciences, Prague, Czech Republic  J. Vink
Universiteit van Amsterdam, Faculty of Science, Amsterdam, The Netherlands  S.
Vorobiov  Center for Astrophysics and Cosmology (CAC), University of Nova
Gorica, Nova Gorica, Slovenia  H. Wahlberg  IFLP, Universidad Nacional de La
Plata and CONICET, La Plata, Argentina  C. Watanabe  Universidade Federal do
Rio de Janeiro, Instituto de Física, Rio de Janeiro, RJ, Brazil  A.A. Watson
School of Physics and Astronomy, University of Leeds, Leeds, United Kingdom
A. Weindl  Karlsruhe Institute of Technology (KIT), Institute for
Astroparticle Physics, Karlsruhe, Germany  L. Wiencke  Colorado School of
Mines, Golden, CO, USA  H. Wilczyński  Institute of Nuclear Physics PAN,
Krakow, Poland  D. Wittkowski  Bergische Universität Wuppertal, Department of
Physics, Wuppertal, Germany  B. Wundheiler  Instituto de Tecnologías en
Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina
A. Yushkov  Institute of Physics of the Czech Academy of Sciences, Prague,
Czech Republic  O. Zapparrata  Université Libre de Bruxelles (ULB), Brussels,
Belgium  E. Zas  Instituto Galego de Física de Altas Enerxías (IGFAE),
Universidade de Santiago de Compostela, Santiago de Compostela, Spain  D.
Zavrtanik  Center for Astrophysics and Cosmology (CAC), University of Nova
Gorica, Nova Gorica, Slovenia  Experimental Particle Physics Department, J.
Stefan Institute, Ljubljana, Slovenia  M. Zavrtanik  Experimental Particle
Physics Department, J. Stefan Institute, Ljubljana, Slovenia  Center for
Astrophysics and Cosmology (CAC), University of Nova Gorica, Nova Gorica,
Slovenia  L. Zehrer  Center for Astrophysics and Cosmology (CAC), University
of Nova Gorica, Nova Gorica, Slovenia<EMAIL_ADDRESS>http://www.auger.org

(August 29, 2024)

###### Abstract

Using the data of the Pierre Auger Observatory, we report on a search for
signatures that would be suggestive of super-heavy particles decaying in the
Galactic halo. From the lack of signal, we present upper limits for different
energy thresholds above ${\gtrsim}10^{8}$ GeV on the secondary by-product
fluxes expected from the decay of the particles. Assuming that the energy
density of these super-heavy particles matches that of dark matter observed
today, we translate the upper bounds on the particle fluxes into tight
constraints on the couplings governing the decay process as a function of the
particle mass. Instantons, which are non-perturbative solutions to Yang-Mills
equations, can give rise to decay channels otherwise forbidden and transform
stable particles into meta-stable ones. Assuming such instanton-induced decay
processes, we derive a bound on the reduced coupling constant of gauge
interactions in the dark sector: $\alpha_{X}\lesssim 0.09$, for
$10^{9}\lesssim M_{X}/\text{GeV}<10^{19}$. Conversely, we obtain that, for
instance, a reduced coupling constant $\alpha_{X}=0.09$ excludes masses
$M_{X}\gtrsim 3\times 10^{13}\leavevmode\nobreak\ $GeV. In the context of dark
matter production from gravitational interactions alone during the reheating
epoch, we derive constraints on the parameter space that involves, in addition
to $M_{X}$ and $\alpha_{X}$, the Hubble rate at the end of inflation, the
reheating efficiency, and the non-minimal coupling of the Higgs with
curvature.

##  I Introduction

The aim of this study is to search for signatures of Planckian-interacting
massive particles in the data of the Pierre Auger Observatory and to derive
constraints on the particle physics and cosmological parameters governing the
viability of the Planckian scenario of dark matter (DM). Ultra-high energy
cosmic rays (UHECRs), those cosmic rays with energies above ${\simeq}10^{8}$
GeV, are charged particles accelerated by electromagnetic fields in special
astrophysical environments. Still, the search for subdominant fluxes of
particles that could reveal either some new mechanism of particle acceleration
or new physics is continuously gaining sensitivity with the increased exposure
of the Pierre Auger Observatory [1]. Should one detect UHECRs, and in
particular photons, clustered preferentially in the direction of the Galactic
Center, then this could provide compelling evidence of the presence of super-
heavy relics produced in the early universe and decaying today [2, 3]. Such
super-heavy particles have been proposed to form the DM [4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16].

The nature of DM remains elusive. The leading benchmark relies on assuming the
existence of weakly-interacting massive particles (WIMPs) that were in
equilibrium in the thermal bath of the early universe before dropping out of
equilibrium when the temperature became lower than their mass [17, 18, 19]. To
explain the relic abundance of DM observed today, the mass of these particles
should lie in the range $10^{2}$–$10^{4}$ GeV, which is consistent with the
expectations from the technical naturalness to have new physics at the TeV
scale [20]. However, WIMPs have escaped any detection so far [21, 22, 23]. All
in all, the various null results give increasingly strong constraints for the
WIMPs to match the relic density. Although the exploration of the complete
WIMP parameter space remains of great importance, the current lack of signal
provides a motivation to consider alternative models of DM.

There are good motives for considering super-heavy DM (SHDM) particles rather
than WIMPs. New physics could manifest only at a very high energy scale, such
as the GUT scale ($M_{\text{GUT}}$) or even the Planck scale
($M_{\text{Pl}}$). Such a possibility has emerged from the estimation of the
instability scale $\Lambda_{\text{I}}$ of the Standard Model (SM) that
characterizes the scale at which the SM Higgs potential develops an
instability at large field values. For the current values of the Higgs and top
masses and the strong coupling constant, the range of $\Lambda_{\text{I}}$
turns out to be high, namely $10^{10}$ to $10^{12}$ GeV [24, 25, 26]. While
the change of sign of the Higgs quartic coupling $\lambda$ at that scale could
trigger a vacuum instability due to the Higgs potential suddenly becoming
unbounded from below, the running of $\lambda$ for energies above
$\Lambda_{\text{I}}$ turns out to be slow [24]. This peculiar behaviour leaves
the possibility of extrapolating the SM to even higher energies than
$\Lambda_{\text{I}}$, up to $M_{\text{Pl}}$, with no need to introduce new
physics to stabilize the SM. In this case, the mass spectrum of the dark
sector could reflect the high energy scale of the new physics.

Various mechanisms taking place at the end of the inflationary era in Big Bang
cosmology are capable of producing SHDM particles. Inflation could be driven
by the presence of a scalar field, the inflaton, which slowly rolled down its
potential during the inflationary era before reaching its minimum. The
inflaton field then started coherent oscillations around its minimum potential
and subsequently decayed into SM particles that reheated the universe (the
reheating era) while thermalizing. The production of SHDM could have occurred
in the same manner on the condition that the inflaton experienced a steep
potential right after the period of slow-rolling motion so as to generate
large-amplitude oscillations (see, e.g., [27]). The coupling between the
inflaton and the particles is however required to be fine-tuned to a very
small value to avoid overshooting the DM content. Alternatively, SHDM could
also be produced during the coherent oscillations of the inflaton prior to its
decay, due to the “non-adiabatic” expansion of the background space-time
acting on the vacuum quantum fluctuations [28, 8]. Particles with masses of
the order of the inflaton mass can result from this gravitational production
mechanism. Constraints on such scenarios have already been placed using
cosmic-ray data at ultra-high energies [29], and will be updated and
complemented in a forthcoming publication. In this article and the
accompanying Letter [30], we instead consider particles with masses anywhere
between ${\simeq}10^{8}$ GeV and $M_{\text{Pl}}$. These can have been produced
after the period of inflation has ended by annihilation of SM particles
through the exchange of a graviton [11], or by annihilation of inflaton
particles through the same exchange of a graviton [16]. In this context, the
only interaction between SM and dark sectors is gravitational. For this
reason, these SHDM particles have been dubbed as Planckian-interacting massive
particles (PIDM), and we shall use this term hereafter when we need to be
specific to this minimal coupling for SHDM particles – keeping the term SHDM
for setups with additional feeble couplings. The absence of DM-SM couplings is
consistent with the large panoply of observational evidence for the existence
of DM based on gravitational effects alone. Once SM and inflaton particles
have populated the dark sector prior to the radiation-dominated era, the
abundance of PIDM particles set by the freeze-in mechanism [31, 32, 33] can
evolve to match the relic abundance of DM inferred today for viable parameters
governing the thermal history and geometry of the universe [11].

The absence of direct coupling between PIDM and the SM (apart from
gravitational) leaves only a few possible observational signatures. The large
values of the Hubble expansion rate at the end of inflation $H_{\text{inf}}$
needed to match the relic abundance $\Omega_{\text{CDM}}h$ imply tensor modes
in the cosmological microwave background anisotropies that could be observed
in the future [11]. On the other hand, even if the absence of direct
interactions guarantees the stability of the particles in the perturbative
domain, PIDM protected from decay by a symmetry can eventually disintegrate
due to non-perturbative effects in non-abelian gauge theories and produce
UHECRs such as (anti-)protons/neutrons, photons and (anti-)neutrinos. The aim
of this study is to search for such signatures in the data from the Pierre
Auger Observatory and to derive constraints on the various particle-physics
and cosmological parameters governing the viability of the PIDM scenario for
DM.

The paper is organized as follows. In section II, we derive upper limits on
the flux of secondary by-products expected from the decay of the particles. We
show in particular that the most stringent limits are provided by the absence
of UHE photons. By relating, in the framework of instanton-induced decay, the
lifetime of the particles to the coupling constant $\alpha_{X}$ of a hidden
sector pertaining to PIDM, the limits obtained in section II are shown in
section III to be sufficient to provide upper bounds on $\alpha_{X}$ as a
function of $M_{X}$. Here $\alpha_{X}$ is the gauge coupling constant of a
hidden non-abelian symmetry possibly unified with SM interactions at a high
scale. In section IV, we use the results obtained in [11, 16] for the PIDM
scenario to relate the reheating temperature $T_{\text{rh}}$ (the temperature
at the end of the reheating era), the Hubble expansion rate $H_{\text{inf}}$
and the mass of the particles $M_{X}$ to the relic abundance
$\Omega_{\text{CDM}}h=(0.1199\pm 0.0022)$ [34], with $h$ being the
dimensionless Hubble constant [34]. The relationship obtained is then used to
delineate viable regions to these quantities and $\alpha_{X}$. In parallel, it
is important to assess the possible impacts of inflationary cosmologies on the
astronomically-long lifetime of the vacuum of the SM [24, 35]. Large
fluctuations of free fields generated by the dynamics on a curved background,
because of the presence of a non-minimal coupling $\xi$ between the Higgs
field and the curvature of space-time, might indeed challenge this lifetime.
Requiring the electroweak vacuum not to decay yields constraints between the
non-minimal coupling and the Hubble rate at the end of inflation [36], which
are propagated in the plane $(\xi,\alpha_{X})$ in section V. Finally, the
results are summarized in section VI.

##  II Searches for SHDM/PIDM signatures at the Pierre Auger Observatory

Regardless of the underlying model of particle physics that regulates the
decay process of the SHDM particles, pairs of quarks and anti-quarks of any
flavor are expected as by-products of disintegration. They give rise to a
direct production of fluxes of UHE photons and neutrinos as well as to a
cascade of partons that then produce a cascade of hadrons, among which are
nucleons and pions, which themselves decay and generate copious fluxes of UHE
photons and neutrinos. All these secondaries can be scrutinized in UHECR data.

###  II.1 Prediction of the fluxes of secondaries

Secondaries are expected to be emitted isotropically, in proportion to the DM
density accumulated in galaxy halos. For each particle
$i=\\{\gamma,\nu,\overline{\nu},N,\overline{N}\\}$, the flux as observed on
Earth is dominated by the contribution of the Milky Way halo. It can be
obtained by integrating the position-dependent emission rate $q_{i}$ per unit
volume and unit energy along the path in the direction $\mathbf{n}$,

$J_{i}(E,\mathbf{n})=\frac{1}{4\pi}\int_{0}^{\infty}\mathrm{d}s\leavevmode\nobreak\
q_{i}(E,\mathbf{x}_{\odot}+\mathbf{x}_{i}(s;\mathbf{n})).$ (1)

Here, $\mathbf{x}_{\odot}$ is the position of the Solar system in the Galaxy,
$s$ is the distance from $\mathbf{x}_{\odot}$ to the emission point, and
$\mathbf{n}\equiv\mathbf{n}(\ell,b)$ is a unit vector on the sphere pointing
to the longitude $\ell$ and latitude $b$, in Galactic coordinates. The $4\pi$
normalisation factor accounts for the isotropy of the decay processes.

Figure 1: Energy spectra of decay by-products of an SHDM particle
($M_{X}=M_{\text{Pl}}$ here) in the $q\bar{q}$ channel, based on the
hadronization process described in [37].

The emission rate is shaped by the DM density $n_{\text{DM}}$, more
conveniently expressed in terms of energy density
$\rho_{\text{DM}}=M_{X}\,n_{\text{DM}}$, and by the differential decay width
into the particle species $i$ as

$q_{i}(E,\mathbf{x})=\frac{\rho_{\text{DM}}(\mathbf{x})}{M_{X}}\frac{\mathrm{d}\Gamma_{i}(E;M_{X})}{\mathrm{d}E}.$
(2)

The ingredients are thus well separated in terms of astrophysical and
particle-physics inputs. There are uncertainties in the determination of the
profile $\rho_{\text{DM}}$. We use here the traditional NFW profile as a
reference [38],

$\rho_{\text{DM}}(R)=\frac{\rho_{s}}{(R/R_{s})(1+R/R_{s})^{2}},$ (3)

where $R$ is the distance to the Galactic center, $R_{s}=24$ kpc, and
$\rho_{s}$ is fixed by the DM density in the solar neighborhood, namely
$\rho_{\odot}=0.3$ GeV cm-3. There are uncertainties in the determination of
this profile. We will use other profiles such as those from Einasto [39],
Burkert [40] or Moore [41] as sources of systematics. The other ingredient
shaping the emission rate is the particle-physics factor that regulates the
fluxes of secondary UHECRs from the decay of the super-heavy particles. In
most of SHDM models, the decay is assumed to occur initially in the
parton/anti-parton channel (refereed to as $q\overline{q}$ channel). The
factor is then the (inclusive) differential decay width into secondary $i$
that accounts for the parton cascade and hadronization process. For a particle
with mass $M_{X}$ decaying into partons $a$ that hadronize into particles of
type $h$, the differential width $\mathrm{d}\Gamma_{i}/\mathrm{d}E$ relies
primarily on the hadron energy spectrum, which can be written as [42]

$\frac{\mathrm{d}N_{h}(x,M^{2},M_{X}^{2})}{\mathrm{d}x}=\sum_{a}\int_{x}^{1}\frac{\mathrm{d}z}{z}\frac{1}{\Gamma_{a}}\frac{\mathrm{d}\Gamma_{a}(y,M_{X}^{2})}{\mathrm{d}y}\Bigg{|}_{y=x/z}D_{a}^{h}(z,M^{2}).$
(4)

Here, $x=2E_{h}/M_{X}$, $z=E_{h}/E_{a}$ and $y=x/z$ are the various fractions
of available maximum momentum and primary parton momentum carried by the
hadron under scrutiny. To lowest order for a two-body decay, the decay width
of the particle into parton $a$, $\mathrm{d}\Gamma_{a}/\mathrm{d}y$, is
proportional to $\delta(1-y)$, so that $\mathrm{d}N_{h}/\mathrm{d}x$ is then
proportional to $\sum_{a}D_{a}^{h}(x,M^{2})$, the constant of proportionality
being the inverse of the number of quark flavors $n_{\mathrm{F}}$ [43]. The
$D_{a}^{h}(z,M^{2})$ functions are the fragmentation functions for hadrons of
type $h$ from partons $a$, with $M^{2}$ the factorisation scale chosen to be
$M^{2}\simeq M_{X}^{2}$. These functions are evolved, starting from
measurements at the electroweak scale up to the energy scale fixed by $M_{X}$,
using the DGLAP equation to account for the splitting function that describes
the emission of parton $k$ by parton $j$. The energy spectra of photons,
neutrinos and nucleons, $\mathrm{d}N_{i}/\mathrm{d}x$ with
$i=\\{\gamma,\nu,N\\}$, then follow from the subsequent decay of unstable
hadrons. Among the various computational schemes [44, 45, 37, 46, 47], there
is a general agreement for these spectra to be of the form $E^{-1.9}$. We use
the scheme of Ref. [37] in this study, which is illustrated for the
quark/anti-quark channel in Fig. 1 in terms of $\mathrm{d}N_{i}/\mathrm{d}x$.
Note that to study decays into $p$ quarks/anti-quarks pairs ($p>1)$, the phase
space factor entering into Eq. (4) through the width
$\mathrm{d}\Gamma_{a}/\mathrm{d}y$ then scales as $(2p-1)(2p-2)z(1-z)^{2p-3}$
[44].

All in all, this allows us to express $q_{i}$ as

$q_{i}(E,\mathbf{x})=\frac{\rho_{\text{DM}}(\mathbf{x})}{M_{X}\tau_{X}}\frac{\mathrm{d}N_{i}(E;M_{X})}{\mathrm{d}E},$
(5)

with $\tau_{X}=\Gamma^{-1}_{X}$ the lifetime of the $X$ particles. The salient
features of the flux from the decay by-products of super-heavy particles are
thus the presence of 2-to-3 (3-to-4) times more photons (neutrinos) than
nucleons on the one hand, and its peculiar directional dependency.

###  II.2 Search for secondaries from the decay of SHDM in data of the
Observatory

Figure 2: Signal term of the directional density,
$\delta\mu(\mathbf{n},E=32\,\text{EeV})$, as expected to be observed at the
Pierre Auger Observatory in galactic coordinates.

The features described above can give rise to observational signatures that
can be captured at the Pierre Auger Observatory, located in the province of
Mendoza (Argentina) and covering 3000 km2 [1]. UHECRs can only be studied
through the detection of the showers of particles they create in the
atmosphere. As the cascade develops, nitrogen and oxygen molecules get excited
by the many ionizing electrons created along the shower track. The ultraviolet
fluorescence caused by the subsequent de-excitation of the molecules can be
detected by telescope stations, made up of arrays of several hundreds of
photomultiplier tubes that, thanks to a set of mirrors, each monitor a small
portion of the sky. The isotropic emission enables observing the cascades
side-on up to 30 or 40 km away on moonless nights and thus the reconstruction
of the longitudinal profile of the showers. This reconstruction allows the
inference of both the energy of the showers in a calorimetric way, without
recourse to external information to calibrate the energy estimator, and the
slant depth of maximum of shower development, ($X_{\text{max}}$), which is a
proxy, the best available to now, of the primary mass of the particles.
Complementing the fluorescence detectors, particle detectors deployed on the
ground can be operated with a quasi-permanent duty cycle and thus provide a
harvest of data. The subset of events detected simultaneously by the
fluorescence and the surface detectors is used to develop a calibration curve
such that an energy estimate can be assigned to each event [48, 49, 50]. Such
a hybrid-detection approach is advantageous for providing a calorimetric
estimate of the energy for events recorded during periods when the telescopes
cannot be operated, thus avoiding assumptions about the primary mass and the
hadronic processes that control the shower development to infer the energies.

The Pierre Auger Observatory is such a hybrid system. The array of particle
detectors is made of 1600 water-Cherenkov detectors deployed on a 1500 m
triangular grid. The array is overlooked from four stations, each containing
six telescopes used to detect the emitted fluorescence light. The energy
resolution achieved is 10% above $10^{10}$ GeV [48]. The integrated exposure
of the Observatory over the last 17 years, 122 000 km2 sr yr, has enabled us
to measure the arrival directions, within 1∘ [51], of more than 2 600 UHECRs
above $3.2{\times}10^{10}$ GeV. This data set, the largest available at such
energies, is used to search for a component of UHECRs following the arrival
direction pattern predicted by Equation (1). Previous related searches have
been conducted using much more modest data sets [52, 53, 54, 55, 56, 57]. The
high energy thresholds considered here, namely from $10^{10.5}$ GeV to
$10^{10.9}$ GeV, allow us to minimize the uncertainties inherent in the
modelling of the Galactic magnetic field bending the (anti-)proton
trajectories. A thorough exploration of the entire energy range accessible to
the Observatory is left for a future study.

To search for a sub-dominant directional dependency suggestive of a DM signal,
the set of observed arrival directions is required to match in the best
possible way a directional density
$\mu(\mathbf{n},E)\equiv\mu(\mathbf{n},{>}E)$ that consists of the sum of a
background density and a signal density built from Eq. (1). The balance
between the two contributions is left free and denoted as $\zeta$. As the
dependencies with energy of the background and of the signal terms are
different, the resolution effects (in energy) are expected to distort the
balance parameter. A forward-folding of the detector effects is thus carried
out to build $\mu(\mathbf{n},E;\zeta)$. Under these conditions, the isotropic
background density above an energy threshold $E$,
$\mu_{\text{bkg}}(\mathbf{n},E;\zeta)$, is modelled as

$\mu_{\text{bkg}}(\mathbf{n},E;\zeta)=\omega(\mathbf{n})\int_{{>}E}\mathrm{d}E^{\prime}\int\mathrm{d}E_{0}\leavevmode\nobreak\
J_{\mathrm{bkg}}(E_{0};\zeta)\leavevmode\nobreak\
\kappa_{\text{bkg}}(E^{\prime},E_{0}),$ (6)

where $\omega(\mathbf{n})$ is the directional exposure [58],
$J_{\text{bkg}}(E_{0};\zeta)$ is the energy spectrum of the background built
such that the total energy spectrum $J(E)$ reported in Ref. [48] is the sum of
the background and the signal contributions,

$J_{\text{bkg}}(E_{0};\zeta)=J(E)-\frac{\zeta}{4\pi}\int\mathrm{d}\mathbf{n}\leavevmode\nobreak\
\sum_{i}J_{i}(E,\mathbf{n}),$ (7)

and $\kappa_{\text{bkg}}(E^{\prime},E_{0})$ is the response function of the
detector. In the energy range of interest, the latter reduces to a pure
resolution function [48]. The signal term, on the other hand, is given by

$\delta\mu(\mathbf{n},E;\zeta)=\zeta\omega(\mathbf{n})\int_{{>}E}\mathrm{d}E^{\prime}\int\mathrm{d}E_{0}\leavevmode\nobreak\
\sum_{i}J_{i}(E_{0},\mathbf{n})\kappa_{i}(E^{\prime},E_{0}).$ (8)

Both the response function and the “lookback position” of the particles in the
Galaxy detected in the direction $\mathbf{n}$, $\mathbf{x}_{i}(s;\mathbf{n})$,
depend on the nature of the particles:

  * •

photons: a resolution function $\kappa_{\gamma}$ accounts for a bias (factor 2
at 30 EeV decreasing smoothly to 1 at 100 EeV) [59], while the lookback
position is via straight-line motion, $\mathbf{x}_{\text{n}}(s)=s\mathbf{n}$.

  * •

(anti-)neutrons: the resolution function is approximated by that of the
background, $\kappa_{\text{n}}=\kappa_{\text{bkg}}$, while the lookback
position is via straight-line motion, $\mathbf{x}_{\text{n}}(s)=s\mathbf{n}$.
The attenuation is neglected given the large decay-length value in the energy
range scrutinized.

  * •

(anti-)protons: the resolution function is approximated by that of the
background, $\kappa_{\text{p}}=\kappa_{\text{bkg}}$, while the lookback
position is using the well-established method that consists of retro-
propagating protons and anti-protons from the Earth, counting the time spent
in $\rho_{\text{DM}}$ before exiting the Galaxy [60]. The magnetic field model
contains the so-called JF12 regular component [61] and a turbulent one, the
amplitude of which is fixed to equal the envelope of the regular field.

  * •

(anti-)neutrinos: they are not accounted for in this anisotropy-search
analysis, given the absence of a contribution to the observed number of
events.

The resulting density $\delta\mu(\mathbf{n},E)$ is shown in Fig. 2 for $E=32$
EeV. The final density fitted to the data through a likelihood function
$L(\zeta)=\prod_{\text{events}}\mu(\mathbf{n}_{i},E;\zeta)$ is normalised to 1
when integrated over arrival directions,

$\mu(\mathbf{n},E;\zeta)=\frac{\mu_{\text{bkg}}(\mathbf{n},E;\zeta)+\delta\mu(\mathbf{n},E;\zeta)}{\int\mathrm{d}\mathbf{n}\,\mu_{0}(\mathbf{n},E;\zeta)+\int\mathrm{d}\mathbf{n}\,\delta\mu(\mathbf{n},E;\zeta)}.$
(9)

The analysis is performed for energy thresholds spaced by $\Delta\lg{E}=0.1$.
The largest deviation from the no-signal hypothesis is insignificant (within
$2\sigma$) for $\lg{(E/\text{GeV})}=10.7$. Upper limits at $90\%$ C.L. on the
all-sky-averaged $J_{\text{DM}}(E)\equiv\sum_{i}J_{i}(E)$ flux are then
obtained by solving with Monte Carlo simulations the equation
$\int_{{\geq}\mathcal{L}_{\text{data}}}\mathrm{d}\mathcal{L}\leavevmode\nobreak\
p(\mathcal{L}(\zeta_{90}))=0.90$ and are reported as the red filled circles in
Fig. 3.

Apart from the anisotropies present in the arrival directions, another
signature in favor of the decay of SHDM particles would be the presence of UHE
photons in the data of the Observatory. The identification of photon primaries
relies on the ability to distinguish the showers generated by photons from
those initiated by the overwhelming background of nuclei. Since the radiation
length in the atmosphere is more than two orders of magnitude smaller than the
mean free path for photo-nuclear interactions, the transfer of energy to the
hadron/muon channel in photon showers is reduced with respect to the bulk of
hadron-induced showers, resulting in a lower number of secondary muons.
Additionally, as the development of photon showers is delayed by the typically
small multiplicity of electromagnetic interactions, they reach
$X_{\text{max}}$ deeper in the atmosphere with respect to showers initiated by
hadrons. Both the ground signal and $X_{\text{max}}$ can be measured at the
Observatory. Although showers are observed at a fixed slice in depth with the
array of particle detectors, the longitudinal development is embedded in the
signals detected. The fluorescence and particle detectors are complemented
with the low-energy enhancements of the Observatory, namely three additional
fluorescence telescopes with an elevated field of view, overlooking a denser
array of particle detectors, in which the stations are separated by 750 m. The
combination of these instruments allows showers to be measured in the energy
range above $10^{8}$ GeV.

Figure 3: Upper limits on secondaries produced from the decay of SHDM
particles.

Three different analyses, differing in the detector used, have been developed
to cover the wide energy range probed at the Observatory and have been
reported in Ref. [62, 63, 64]. No photons with energies above
$2{\times}10^{8}$ GeV have been unambiguously identified so far, leading to
the 95% C.L. flux upper limits displayed in Fig. 3 as the filled blue squares.
The limit above $10^{11.2}$ GeV (green triangle), stemming from the non-
detection so far of any UHECR [48], including photons, is also constraining
[47, 65]. In the energy range above $2{\times}10^{10}$ GeV, the limits on
photon fluxes are observed to be much more constraining than those inferred
from the absence of significant anisotropies. This is because the accumulated
exposure to photons enables us to probe fluxes less than a few percent of that
of UHECRs, while the current sensitivity to anisotropies does not allow for
capturing an amplitude less than 10 to 15% of the UHECR flux.

Finally, (anti-)neutrinos, another emblematic signature of SHDM particle
decays, can also be identified at the Observatory. Neutrinos of all flavors
can interact in the atmosphere through charged- or neutral-current
interactions and induce a “downward-going” shower that can be detected [66].
In addition, tau neutrinos ($\nu_{\tau}$) can undergo charged-current
interactions and produce a $\tau$ lepton in the Earth’s crust that eventually
decays in the atmosphere, inducing an upward-going shower [67]. Tau neutrinos
are not expected to be copiously produced at the astrophysical sources; yet
approximately equal fluxes for each neutrino flavour should reach the Earth as
a result of neutrino oscillations over cosmological distances [68, 69, 70].
The identification of neutrinos relies on salient zenith-dependent features of
air showers. For highly-inclined cascades (zenith angle larger than 60∘),
neutrino-induced showers initiated deep in the atmosphere near ground level
have a significant electromagnetic component when they reach the array of
particle detectors, producing signals that are spread over time. In contrast,
inclined showers initiated at a shallow depth in the atmosphere by the bulk of
UHECRs are dominated by muons at the ground level, inducing signals in the
particle detectors that have characteristic high peaks associated with
individual muons, which are spread over smaller time intervals. Thanks to the
fast sampling (25 ns) of the digital electronics of the detectors, several
observables that are sensitive to the time structure of the signal can be used
to discriminate between these two types of showers.

Neutrino limits obtained at the Observatory [71] are also displayed in Fig. 3
as the continuous line. Except at the lowest energies, these limits are seen
to be superseded by photon limits in the search for SHDM by-product decays.

##  III Constraints on gauge coupling in the dark sector

###  III.1 Pertubative-decay processes

Some SHDM models postulate the existence of super-weak couplings between the
dark and SM sectors. The lifetime $\tau_{X}$ of the particles is then governed
by the strength of the couplings $g_{X\Theta}$ (or reduced couplings
$\alpha_{X\Theta}=g_{X\Theta}^{2}/(4\pi)$) and by the mass dimension $n$ of
the operator $\Theta$ standing for the SM fields in the effective interaction
[72]. Even without knowing the theory behind the decay of the DM particle, we
can derive generic constraints on $\alpha_{X\Theta}$ and $n$. The effective
interaction term that couples the field $X$ associated with the heavy particle
to the SM fields is taken as

$\mathcal{L}_{\text{int}}=\frac{g_{X\Theta}}{\Lambda^{n-4}}X\Theta,$ (10)

where $\Lambda$ is an energy parameter typical of the scale of the new
interaction. In the absence of further details about the operator $\Theta$,
the matrix element describing the decay transition is considered flat in all
kinematic variables so that it behaves as $|\mathcal{M}|^{2}\sim
4\pi\alpha_{X\Theta}/\Lambda^{2n-4}$. On the basis of dimensional arguments,
the lifetime of the particle $X$ is then given as

$\tau_{X\Theta}=\frac{V_{n}}{4\pi
M_{X}\alpha_{X\Theta}}\left(\frac{\Lambda}{M_{X}}\right)^{2n-8},$ (11)

where $V_{n}$ is a phase space factor. As a proxy for this factor, we use the
expression derived for $N-1$ particles in the final state [73],

$V_{n}=\left(\frac{2}{\pi}\right)^{n-1}\Gamma(n-1)\Gamma(n-2),$ (12)

with $\Gamma(x)$ the Euler gamma function.

Figure 4: Exclusion regions in the plane $(\alpha_{X\Theta},M_{X})$ for
several values of mass dimension $n$ of operators responsible for the
perturbative decay of the super-heavy particle, and for an energy scale of new
physics $\Lambda=10^{16}$ GeV.

Equation (11) provides us with a relationship connecting the lifetime
$\tau_{X\Theta}$ to the coupling constant $\alpha_{X\Theta}$ and to the
dimension $n$.

From Eq. (11), it is apparent that the coupling constant $\alpha_{X\Theta}$
and the dimension $n$ have to take specific values for super-heavy particles
to be stable enough [4, 72]. We now show that the absence of UHE photons
provides powerful data to infer the viable range of values. Assuming that the
relic abundance of DM is saturated by SHDM, constraints can be inferred in the
plane $(\tau_{X\Theta},M_{X})$ by requiring the flux calculated by averaging
Equation (1) over all directions to be less than the limits,

$\int_{E}^{\infty}\mathrm{d}E^{\prime}\langle
J_{\gamma}(E^{\prime},\mathbf{n})\rangle\leq J^{95\%}_{\gamma}({\geq}E),$ (13)

where $\langle\cdot\rangle$ stands for the average over all directions. In
practice, for a specific upper limit at one energy threshold, a lower limit of
the $\tau_{X\Theta}$ parameter is derived for each value of mass $M_{X}$. The
lower limit on $\tau_{X\Theta}$ is subsequently transformed into an upper
limit on $\alpha_{X\Theta}$ by means of Eq. (11). This defines a curve in the
plane $(\tau_{X\Theta},M_{X})$. By repeating the procedure for each upper
limit on $J^{95\%}_{\gamma}(\geq E)$, a set of curves is obtained, reflecting
the sensitivity of a specific energy threshold to some range of mass. The
union of the excluded regions finally provides the constraints in the plane
$(\alpha_{X\Theta},M_{X})$. In this manner we obtain the contour lines shown
in Fig. 4 for several values of $n$ and for an emblematic choice of GUT
$\Lambda$ value. The scale chosen for $\alpha_{X\Theta}$ ranges from 1 down to
$10^{-5}$. It is observed that for the limits on photon fluxes to be
satisfied, the mass of the super-heavy particle cannot exceed
${\gtrsim}10^{9}$ GeV (${\gtrsim}10^{11}$ GeV) for operators of dimension
equal to or larger than $n=8$ ($n=10$), while larger masses require an
increase in $n$. To approach the large masses while keeping operators of
dimension relatively low, “astronomically-small” coupling constants should be
at work. The same conclusions hold for other choices of $\Lambda$. All in all,
for perturbative processes to be responsible for the decay of SHDM particles
requires quite “unnatural” fine-tuning.111See, however, Ref. [74] for a model
in which SHDM couples to the neutrino sector.

###  III.2 Instanton-induced decay processes

The sufficient stability of super-heavy particles is better ensured by a new
quantum conserved in the dark sector so as to protect the particles from
decaying. The only interaction between the dark sector and the SM one is then
gravitational, as in the PIDM instance of SHDM models. Nevertheless, even
stable particles in the perturbative domain will in general eventually decay
due to non-perturbative effects in non-abelian gauge theories. Such effects,
known as instantons [75, 76, 77], provide a signal for the occurrence of
quantum tunneling between distinct classes of vacua, forcing the fermion
fields to evolve during the transitions and leading to the generation of
particles depending on the associated anomalous symmetries [78]. Instanton-
induced decay can thus make observable a dark sector of PIDM particles that
would otherwise be totally hidden by the conservation of a quantum number.
Following Ref. [79], we assume quarks and leptons carry this quantum number
and so contribute to anomaly relationships with contributions from the dark
sector,222Alternatively, the particles of the dark sector could carry some SM
hypercharge. they will be secondary products in the decays of PIDM together
with the lightest hidden fermion. The presence of quarks and leptons in the
final state is sufficient to make usable the hadronization process described
in Section II. The exact particle content is governed by selection rules
arising from the instanton transitions that are regulated by the fermions
coupled to the gauge field of the dark sector. As a proxy inspired from Ref.
[79], we assume here that a dozen of $q\bar{q}$ pairs are produced in the
decay process and that half of the energy goes into the dark sector.

Figure 5: Upper limits at 95% C.L. on the coupling constant $\alpha_{X}$ of a
hidden gauge interaction as a function of the mass $M_{X}$ of a dark matter
particle $X$ decaying into a dozen of $q\bar{q}$ pairs. The dotted and dashed-
dotted lines represent the systematic uncertainties stemming from the quantum
fluctuations about the instanton contribution to the transition amplitude (see
text). For reference, the unification of the three SM gauge couplings is shown
as the horizontal blue dashed line in the framework of supersymmetric GUT
[80].

The lifetime of the decaying particle follows from the corresponding
instanton-transition amplitude obtained as a semi-classical expansion of the
associated path integral about the instanton solution, which provides the
zeroth-order contribution that depends exponentially on $g_{X}^{-2}$ [78]. It
is the introduction of this exponential factor in the effective interaction
term that suppresses to a large extent the fast decay of the particles.
Considering this zeroth-order contribution only, and recasting the expression
in terms of the reduced coupling constant of the hidden gauge interaction
$\alpha_{X}$, the lifetime of the particles is given as

$\tau_{X}\simeq M_{X}^{-1}\exp{\left(4\pi/\alpha_{X}\right)}.$ (14)

In this expression, we dropped, following Ref. [79], the functional
determinants arising from the effect of quantum fluctuations around the
(classical) contribution of the instanton configurations. Those from the Yang-
Mills gauge fields yield a dependency in $(4\pi\alpha_{X})^{5+n_{1}}$ in Eq.
(14) with $n_{1}=3\leavevmode\nobreak\ (7)$ for $SU$(2) ($SU$(3)) theories for
instance, a dependency that is negligible compared to the exponential one in
$\alpha_{X}^{-1}$. Other functional determinants arise from the exact content
of fields of the underlying theory. Again, the constraints inferred on
$\alpha_{X}$ using Eq. (14) are barely changed for a wide range of numerical
factors given the exponential dependency in $\alpha_{X}^{-1}$.

Eq. (14) provides us with a relationship connecting the lifetime $\tau_{X}$ to
the coupling constant $\alpha_{X}$. In the same way as in the perturbative
case above, upper limits on $\alpha_{X}$ can be obtained. They are shown as
the shaded red area in Fig. 5. Our results show that the coupling should be
less than $\simeq 0.09$ for a wide range of masses. As already stated,
numerical factors could however arise in Equation (14) depending on the
underlying model for the hidden gauge sector. For example, for a theory with a
hidden Higgs field responsible for mass generation in the dark sector, the
factors would involve the ratio between the mass of the lightest dark state
and the energy scale of new physics through the vacuum expectation value [81].
Such explicit constructions of the dark sector are, however, well beyond the
scope of this experimental study. Although the limits presented in Fig. 5 are
hardly destabilized due to the exponential dependence in $\alpha_{X}^{-1}$, we
note that a shift of $\pm 0.0013k$ for factors $10^{\pm k}$ and limit
ourselves to showing in dotted and dashed lines the bounds that would be
obtained for $k=2$ and $k=4$, respectively. These factors are by far the
dominant systematic uncertainties.

##  IV Constraints on the production of PIDM particles during reheating

We now turn to the connection between the results presented in Fig. 5 and the
scenarios of inflationary cosmologies. In addition to the instanton-mediated
decays, PIDM particles can interact gravitationally. Two recent studies [11,
16] have shown that the gravitational interaction alone may have been
sufficient to produce the right amount of DM particles at the end of the
inflation era for a wide range of high masses, up to $M_{\text{GUT}}$. PIDM
particles are naturally part of this scheme. While the observation of UHE
photons could open a window to explore high-energy gauge interactions and
possibly GUTs effective in the early universe, the constraints inferred on
$\alpha_{X}$ allow us to probe the gravitational production of PIDM. We give
below the main steps to derive an expression (Eq. (19)) relating the present-
day relic abundance of DM to the mass $M_{X}$ and other relevant parameters;
more details can be found in Refs. [11] and [16].

PIDM particles are assumed to be produced by annihilation of SM particles [11]
or of inflaton particles [16] through the exchange of a graviton after the
period of inflation has ended at time $H_{\text{inf}}^{-1}$. In this context,
SM particles are created by the decay of coherent oscillations of the inflaton
field, $\phi$, with width $\Gamma_{\phi}$, which is regulated by the coupling
of the inflaton to SM particles $g_{\phi}$ and its mass $M_{\phi}$ as
$\Gamma_{\phi}=g_{\phi}^{2}M_{\phi}/(8\pi)$. They subsequently scatter and
thermalize until the reheating era ends at time $\Gamma_{\phi}^{-1}$ when the
radiation-dominated era begins with temperature $T_{\text{rh}}$. This latter
parameter, given by

$T_{\text{rh}}\simeq 0.25\epsilon(M_{\text{Pl}}H_{\text{inf}})^{1/2}$ (15)

with $\epsilon=(\Gamma_{\phi}/H_{\text{inf}})^{1/2}$ the efficiency of
reheating, is obtained by assuming an instantaneous conversion of the energy
density of the inflaton into radiation for a value of the cosmological scale
factor $a$ such that the expansion rate $H_{\text{inf}}$ equates with the
decay width $\Gamma_{\phi}$ [82]. Here, the number of degrees of freedom at
reheating has been assumed to be that of the SM. For an instantaneous
reheating to be effectively achieved, $\epsilon$ must approach 1, which, from
the expression of $\Gamma_{\phi}$, requires $M_{\phi}$ to be of order of
$H_{\text{inf}}$ and $g_{\phi}$ not too weak. In the following, both
$H_{\text{inf}}$ and $\epsilon$ will be considered as free parameters to be
constrained.

The dynamics of the reheating period are quite involved [27, 83].333Note that
we consider throughout this section, as in [27, 83], an equation of state
$w=0$ for the inflaton field dynamics. As the SM particles thermalize, the
plasma temperature rises rapidly to a maximum before subsequently decreasing
as $T(a)\propto a^{-3/8}$,

$T(a)\simeq 0.2(\epsilon
M_{\text{Pl}}H_{\text{inf}})^{1/2}\left(a^{-3/2}-a^{-4}\right)^{1/4}.$ (16)

The $a^{-3/8}$ scaling continues until the age of the universe is equal to
$\Gamma_{\phi}^{-1}$, signaling the beginning of the radiation-dominated era
at temperature $T_{\text{rh}}$. During this period, the Hubble rate $H(a)$
scales as the square root of the energy density of the inflaton,
$\rho_{\phi}$, which itself scales as
$\rho_{\text{inf}}(a_{\text{inf}}/a)^{3}$. Consequently, $H(a)$ evolves as
$a^{-3/2}$, namely $H(a)=H_{\text{inf}}(a/a_{\text{inf}})^{-3/2}$ with
$a_{\text{inf}}$ being the scale factor at the end of inflation. After
reheating, both the temperature and the Hubble rate follow the standard
evolution in a radiation-dominated era, namely $T(a)\propto
T_{\text{rh}}a_{\text{rh}}/a$ and
$H(a)=H_{\text{inf}}\epsilon^{2}(a/a_{\text{rh}})^{-2}$. The scale factor at
the end of reheating is $a_{\text{rh}}=\epsilon^{-4/3}a_{\text{inf}}$,
guaranteeing the continuity of $H(a)$.

Figure 6: Constraints in the $(H_{\text{inf}},M_{X})$ plane. The red region is
excluded by the non-observation of tensor modes in the cosmic microwave
background [11, 84]. The regions of viable $(H_{\text{inf}},M_{X})$ values
needed to set the right abundance of DM are delineated by the curves for
different values of reheating efficiency $\epsilon$ [85] from dark blue
($\epsilon=1$) to lighter ones ($\epsilon=10^{-4}$), while values above
(below) the lines lead to overabundance of (negligible quantity of) DM.
Additional constraints from the non-observation of instanton-induced decay of
SHDM particles allow for excluding the mass ranges in the regions to the right
of the vertical lines, for the specified value of the dark-sector gauge
coupling.

With these reheating dynamics in hand, the relic abundance of PIDM particles
can be estimated. The energy density of the universe is then in the form of
unstable inflaton particles, SM radiation and stable massive particles, the
time evolution of which is governed by a set of coupled Boltzmann equations
[27]. However, because the energy density of the massive particles is always
sub-dominant, the evolution of the inflationary and radiation energy densities
largely decouple from the time evolution of the $X$-particle density $n_{X}$.
In addition, because PIDM particles interact through gravitation only, they
never come to thermal equilibrium. In this case, the collision term in the
Boltzmann equation can be approximated as a source term only,

$\frac{\mathrm{d}n_{X}(t)}{\mathrm{d}t}+3H(t)n_{X}(t)\simeq\sum_{i}\overline{n}_{i}^{2}(t)\gamma_{i}.$
(17)

Here, the sum in the right hand side stands for the contributions from the SM
and inflationary sectors. In the SM sector,
$\overline{n}_{i}=m_{X}^{2}TK_{2}(M_{X}/T)/(2\pi^{2})$ [6], with $K_{2}(x)$
being the modified Bessel function of the second kind, and
$\gamma_{i}=\langle\sigma v\rangle$ is the thermal-averaged cross section
times velocity describing the SM+SM$\to$PIDM+PIDM reaction [11, 85], which
behaves as $M_{X}^{2}/M_{\text{Pl}}^{4}$ for $M_{X}\gg T$ and as
$T^{2}/M_{\text{Pl}}^{4}$ for $M_{X}\ll T$. In the inflationary sector,
$\overline{n}_{i}=\rho_{\text{inf}}(a_{\text{inf}}/a)^{3}/M_{\phi}$, with
$M_{\phi}=3{\times}10^{13}$ GeV in the following, and the production rate
$\gamma_{i}$ describes the $\phi+\phi\to$PIDM+PIDM reaction [16]. In both SM
and inflationary sectors, the production rates $\gamma_{i}$ for fermionic DM
are considered in the following. Introducing the dimensionless abundance
$Y_{X}=n_{X}a^{3}/T_{\text{rh}}^{3}$ to absorb the expansion of the universe,
and using $aH(a)\mathrm{d}t=\mathrm{d}a$ from the definition of the Hubble
parameter, Eq. (17) becomes

$\frac{\mathrm{d}Y_{X}(a)}{\mathrm{d}a}\simeq\frac{a^{2}}{T_{\text{rh}}^{3}H(a)}\sum_{i}\overline{n}_{i}^{2}(a)\gamma_{i},$
(18)

which, using the dynamics of the expansion rate during reheating described
above, yields the present-day dimensionless abundance $Y_{X,0}$ assuming
$Y_{X,\mathrm{inf}}=0$. The present-day relic abundance,
$\Omega_{\text{CDM}}$, can then be related to $M_{X}$, $H_{\text{inf}}$, and
$\epsilon$ through [11]

$\Omega_{\text{CDM}}h^{2}=9.2{\times}10^{24}\frac{\epsilon^{4}M_{X}}{M_{\text{Pl}}}Y_{X,0}.$
(19)

The viable $(H_{\text{inf}},M_{X})$ parameter space is delineated by the
curves corresponding to different values of $\epsilon$ in Fig. 6, from dark
blue ($\epsilon=1$) to lighter ones ($\epsilon=10^{-4}$). As the source term
in the r.h.s. of Eq. (17) raises faster with $H_{\text{inf}}$ than
$T_{\text{rh}}^{3}H(a)$, $Y_{X}$ is a rising function of $H_{\text{inf}}$, and
values for $(H_{\text{inf}},M_{X})$ above (below) the lines lead to
overabundance of (negligible quantity of) DM. For high efficiencies
(corresponding to short duration of the reheating era), the
SM+SM$\to$PIDM+PIDM reaction allows for a wide range of $M_{X}$ values to
fulfill Eq. (19). For $M_{X}$ to be around the GUT scale, the expansion rate
$H_{\text{inf}}$ (being the proxy of the energy scale of the inflation) must
be sufficiently high. Arbitrarily large values of $H_{\text{inf}}$ are however
not permitted because of the 95% C.L. limits on the tensor-to-scalar ratio in
the cosmic microwave background anisotropies, which, once converted into
limits on the energy scale of inflation when the pivot scale exits the Hubble
radius [11, 84], yield $H_{\text{inf}}\leq 4.9{\times}10^{-6}M_{\text{Pl}}$.
For efficiencies below $\simeq 0.01$, the $\phi+\phi\to$PIDM+PIDM reaction
allows for solutions in a narrower range of the $(H_{\text{inf}},M_{X})$
plane, with in particular $M_{X}\leq M_{\phi}$ as a result of the kinematic
suppression in the corresponding rate $\gamma_{i}$ [16].

A clear signature of the PIDM scenario could be the detection of UHE photons
produced by the instanton-induced decay of the PIDM particles – so that no
coupling between the sectors is required except gravitation. The excluded mass
ranges obtained from the non-observation of instanton-induced decay of PIDM
particles are regions to the right of the vertical lines for different values
of dark-sector gauge coupling. While the range of $M_{X}$ extends from (well)
below $10^{8}$ GeV to ${\simeq}10^{17}$ GeV in the case of instantaneous
reheating ($\epsilon=1$) and $\alpha_{X}\leq 0.085$, the parameter space is
observed to shrink for longer reheating duration and larger dark-sector gauge
coupling. With the current sensitivity, there are no longer pairs of values
$(H_{\text{inf}},M_{X})$ satisfying Eq. (19) for $(\epsilon\geq
0.01,\alpha_{X}\geq 0.10)$.

Figure 7: Allowed range of $(\epsilon,\alpha_{X})$ values for the scenario of
Planckian-interactive massive particles as DM delineated for three examples of
$H_{\text{inf}}$. All regions under the lines are excluded.

The allowed range of $(\epsilon,\alpha_{X})$ values is better appreciated in
Fig. 7 for three values of $H_{\text{inf}}$. All regions under the lines are
excluded. For $H_{\text{inf}}=10^{9}$ GeV, the relic density can match the
present-day one provided that $M_{X}$ ranges between ${\simeq}10^{11}$ and
${\simeq}10^{14}$ GeV, $\alpha_{X}$ is less than ${\simeq}0.094$, and the
efficiency $\epsilon$ is larger than ${\simeq}30\%$; otherwise the PIDM
scenario cannot hold to explain the (entire) DM content observed today in the
universe. For larger values of $H_{\text{inf}}$, the allowed range of
$(\epsilon,\alpha_{X})$ gets larger as well as the allowed range of $M_{X}$.
Larger values of $\alpha_{X}$ are possible on the condition of having
$\epsilon$ larger than ${\simeq}\leavevmode\nobreak\ 0.13\%$ (2.7%) for
$H_{\text{inf}}=10^{13}$ GeV ($10^{11}$ GeV). However, note that the available
parameter space shrinks significantly by restricting the allowed mass range to
high values. For the mass of the PIDM particles to lie above $M_{\text{GUT}}$
for instance, the allowed range of $(\epsilon,\alpha_{X})$ values then becomes
$(\geq 0.30,\leq 0.087)$ for $H_{\text{inf}}=10^{13}$ GeV. Probing such a
value of $H_{\text{inf}}$ will be possible with the increased sensitivity to
the tensor-to-scalar ratio through the B-mode polarisation of the cosmic
microwave background anisotropies in the next decade [86, 87]. In parallel,
the sensitivity to UHE photons will also improve thanks to the planned UHECR
observatories [88, 89]. Hence the parameter space allowing for GUT scale
masses will be explored and could be either uncovered or significantly shrunk.

##  V Constraints from SM stability during inflation

As previously stated, the SM Higgs potential develops an instability at large
field value. As a consequence, the SM electroweak vacuum does not correspond
to minimum energy, but to a metastable state. Still, a quantitative estimation
of its rate of quantum tunnelling into a lower energy state in flat space-time
leads to a lifetime comfortably larger than the age of the universe [24, 35].
Such an astronomically-long lifetime is not challenged in the cosmological
context due to thermal fluctuations allowing the decay when the temperature
was high enough [90]. Yet, it might be challenged due to large fluctuations of
free fields generated by the dynamics on a curved background because of the
presence of a non-minimal coupling $\xi$ between the Higgs field and the
curvature of space-time during the inflation period. In such a case, new
degrees of freedom at an intermediate scale below $\Lambda_{\text{I}}$ would
be necessary to stabilize the SM and the PIDM scenario would somehow be
invalidated.

Figure 8: Constraints from vacuum stability in the PIDM scenario. The excluded
ranges of $(\xi,\alpha_{X})$ values for the scenario of Planckian-interactive
massive particles as DM is delineated for three examples of $\epsilon$ – see
text for details. For reference, the non-minimal value for $\xi$ expected from
conformal theories is shown as the dashed line.

The size of the field fluctuations aforementioned are critically determined by
the Hubble rate parameter $H_{\text{inf}}$, which governs the dynamics of the
SM Higgs during inflation. The requirement for the electroweak vacuum to
maintain its astronomical lifetime allows constraints between the non-minimal
coupling $\xi$ and the Hubble rate $H_{\text{inf}}$ in viable regions.
Stability bounds have been derived in the $(\xi,H_{\text{inf}})$ plane by
accounting for the curvature-dependent effective potential of the Higgs up to
one-loop order [36]. They can be propagated into the $(\xi,\alpha_{X})$ plane.
To do so, a scan in the variable $\alpha_{X}$ is performed. For each value of
$\alpha_{X}$, the corresponding upper limit on $M_{X}$ as obtained from Eq.
(14) is used in Eq. (19) to determine the viable $H_{\text{inf}}$ value, which
is finally used to read the associated range of allowed values for $\xi$ from
[36]. We show the result of the analysis in Fig. 8 for three different values
of efficiency. For $\epsilon=1$, the lower-right region delineated by the
black curve is excluded. For $\epsilon=0.1$, the exclusion zone delineated by
the red curve is enlarged. Finally, for $\epsilon=0.01$, the exclusion zone is
delineated by the yellow curve: there are no possible values in the region
$(\xi\lesssim 0.07,\alpha_{X}\gtrsim 0.099)$ for the PIDM scenario to hold.

For reference, the value of $\xi=1/6$ that corresponds to a conformally-
invariant coupling is shown as the dashed line. The experimental bounds from
the LHC are $|\xi|\lesssim 2.6{\times}10^{15}$ [91].

##  VI Conclusion

In this paper, we have considered a class of SHDM scenarios in which the DM
lifetime is stabilized due to having no charges under SM interactions. In this
case, DM may interact with SM particles through instantons of a gauge group
describing the dark sector or only gravitationally. We obtained constraints on
the masses and couplings in such PIDM scenarios by exploiting the limits
placed on the flux of UHE photons using the data of the Pierre Auger
Observatory. In this case, super-heavy particles with masses as large as the
GUT energy scale could be sufficiently abundant to match the DM relic density,
provided that the inflationary energy scale is high ($H_{\text{inf}}\sim
10^{13}$ GeV) and the reheating efficiency is high (so that reheating is
quasi-instantaneous). This rules out values of the dark-sector gauge coupling
greater than ${\simeq}\leavevmode\nobreak\ 0.09$. The mass values could
however be smaller, relaxing the constraints on the efficiency. For more
moderate values of $\epsilon$, the need to avoid more than one bubble
nucleation event in the observable universe during inflation implies then that
the non-minimal coupling of the Higgs to the curvature is more than a few
percent. It is likely that the examples of constraints inferred on models of
dark sectors and physics in the reheating epoch in the framework of
inflationary cosmologies only scratch the surface of the power of limits on
UHE photon fluxes to constrain physics otherwise beyond the reach of
laboratory experiments.

Acknowledgments. The successful installation, commissioning, and operation of
the Pierre Auger Observatory would not have been possible without the strong
commitment and effort from the technical and administrative staff in Malargüe.
We are very grateful to the following agencies and organizations for financial
support: Argentina – Comisión Nacional de Energía Atómica; Agencia Nacional de
Promoción Científica y Tecnológica (ANPCyT); Consejo Nacional de
Investigaciones Científicas y Técnicas (CONICET); Gobierno de la Provincia de
Mendoza; Municipalidad de Malargüe; NDM Holdings and Valle Las Leñas; in
gratitude for their continuing cooperation over land access; Australia – the
Australian Research Council; Belgium – Fonds de la Recherche Scientifique
(FNRS); Research Foundation Flanders (FWO); Brazil – Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq); Financiadora de Estudos e
Projetos (FINEP); Fundação de Amparo à Pesquisa do Estado de Rio de Janeiro
(FAPERJ); São Paulo Research Foundation (FAPESP) Grants No. 2019/10151-2, No.
2010/07359-6 and No. 1999/05404-3; Ministério da Ciência, Tecnologia,
Inovações e Comunicações (MCTIC); Czech Republic – Grant No. MSMT CR LTT18004,
LM2015038, LM2018102, CZ.02.1.01/0.0/0.0/16_013/0001402,
CZ.02.1.01/0.0/0.0/18_046/0016010 and CZ.02.1.01/0.0/0.0/17_049/0008422;
France – Centre de Calcul IN2P3/CNRS; Centre National de la Recherche
Scientifique (CNRS); Conseil Régional Ile-de-France; Département Physique
Nucléaire et Corpusculaire (PNC-IN2P3/CNRS); Département Sciences de l’Univers
(SDU-INSU/CNRS); Institut Lagrange de Paris (ILP) Grant No. LABEX
ANR-10-LABX-63 within the Investissements d’Avenir Programme Grant No.
ANR-11-IDEX-0004-02; Germany – Bundesministerium für Bildung und Forschung
(BMBF); Deutsche Forschungsgemeinschaft (DFG); Finanzministerium Baden-
Württemberg; Helmholtz Alliance for Astroparticle Physics (HAP); Helmholtz-
Gemeinschaft Deutscher Forschungszentren (HGF); Ministerium für Innovation,
Wissenschaft und Forschung des Landes Nordrhein-Westfalen; Ministerium für
Wissenschaft, Forschung und Kunst des Landes Baden-Württemberg; Italy –
Istituto Nazionale di Fisica Nucleare (INFN); Istituto Nazionale di
Astrofisica (INAF); Ministero dell’Istruzione, dell’Universitá e della Ricerca
(MIUR); CETEMPS Center of Excellence; Ministero degli Affari Esteri (MAE);
México – Consejo Nacional de Ciencia y Tecnología (CONACYT) No. 167733;
Universidad Nacional Autónoma de México (UNAM); PAPIIT DGAPA-UNAM; The
Netherlands – Ministry of Education, Culture and Science; Netherlands
Organisation for Scientific Research (NWO); Dutch national e-infrastructure
with the support of SURF Cooperative; Poland – Ministry of Education and
Science, grant No. DIR/WK/2018/11; National Science Centre, Grants No.
2016/22/M/ST9/00198, 2016/23/B/ST9/01635, and 2020/39/B/ST9/01398; Portugal –
Portuguese national funds and FEDER funds within Programa Operacional Factores
de Competitividade through Fundação para a Ciência e a Tecnologia (COMPETE);
Romania – Ministry of Research, Innovation and Digitization, CNCS/CCCDI –
UEFISCDI, projects PN19150201/16N/2019, PN1906010, TE128 and PED289, within
PNCDI III; Slovenia – Slovenian Research Agency, grants P1-0031, P1-0385,
I0-0033, N1-0111; Spain – Ministerio de Economía, Industria y Competitividad
(FPA2017-85114-P and PID2019-104676GB-C32), Xunta de Galicia (ED431C 2017/07),
Junta de Andalucía (SOMM17/6104/UGR, P18-FR-4314) Feder Funds, RENATA Red
Nacional Temática de Astropartículas (FPA2015-68783-REDT) and María de Maeztu
Unit of Excellence (MDM-2016-0692); USA – Department of Energy, Contracts No.
DE-AC02-07CH11359, No. DE-FR02-04ER41300, No. DE-FG02-99ER41107 and No. DE-
SC0011689; National Science Foundation, Grant No. 0450696; The Grainger
Foundation; Marie Curie-IRSES/EPLANET; European Particle Physics Latin
American Network; and UNESCO.

We acknowledge for this work the support of the Institut Pascal at Université
Paris-Saclay during the Paris-Saclay Astroparticle Symposium 2021, with the
support of the P2IO Laboratory of Excellence (program “Investissements
d’avenir” ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038), the P2I axis
of the Graduate School Physics of Université Paris-Saclay, as well as IJCLab,
CEA, IPhT, APPEC, the IN2P3 master projet UCMN and EuCAPT ANR-11-IDEX-0003-01
Paris-Saclay and ANR-10-LABX-0038).

## References

  * Aab  _et al._ [2015] A. Aab _et al._ (Pierre Auger), Nucl. Instrum. Meth. A 798, 172 (2015), arXiv:1502.01323 [astro-ph.IM] .
  * Bhattacharjee and Sigl [2000] P. Bhattacharjee and G. Sigl, Phys. Rept. 327, 109 (2000), arXiv:astro-ph/9811011 .
  * Anchordoqui [2019] L. A. Anchordoqui, Phys. Rept. 801, 1 (2019), arXiv:1807.09645 [astro-ph.HE] .
  * Ellis  _et al._ [1992] J. R. Ellis, G. Gelmini, J. L. Lopez, D. V. Nanopoulos, and S. Sarkar, Nucl. Phys. B 373, 399 (1992).
  * Berezinsky  _et al._ [1997] V. Berezinsky, M. Kachelrieß, and A. Vilenkin, Phys. Rev. Lett. 79, 4302 (1997).
  * Chung  _et al._ [1998] D. J. H. Chung, E. W. Kolb, and A. Riotto, Phys. Rev. D 59, 023501 (1998), arXiv:hep-ph/9802238 .
  * Kuzmin and Tkachev [1999a] V. Kuzmin and I. Tkachev, Phys. Rev. D 59, 123006 (1999a), arXiv:hep-ph/9809547 .
  * Chung  _et al._ [2000] D. J. H. Chung, E. W. Kolb, A. Riotto, and I. I. Tkachev, Phys. Rev. D 62, 043508 (2000), arXiv:hep-ph/9910437 .
  * Kolb  _et al._ [2007] E. W. Kolb, A. A. Starobinsky, and I. I. Tkachev, JCAP 07, 005 (2007), arXiv:hep-th/0702143 .
  * Fedderke  _et al._ [2015] M. A. Fedderke, E. W. Kolb, and M. Wyman, Phys. Rev. D 91, 063505 (2015), arXiv:1409.1584 [astro-ph.CO] .
  * Garny  _et al._ [2016] M. Garny, M. Sandora, and M. S. Sloth, Phys. Rev. Lett. 116, 101302 (2016), arXiv:1511.03278 [hep-ph] .
  * Ellis  _et al._ [2016] J. Ellis, M. A. G. Garcia, D. V. Nanopoulos, K. A. Olive, and M. Peloso, JCAP 03, 008 (2016), arXiv:1512.05701 [astro-ph.CO] .
  * Kolb and Long [2017] E. W. Kolb and A. J. Long, Phys. Rev. D 96, 103540 (2017), arXiv:1708.04293 [astro-ph.CO] .
  * Dudas  _et al._ [2017] E. Dudas, Y. Mambrini, and K. Olive, Phys. Rev. Lett. 119, 051801 (2017), arXiv:1704.03008 [hep-ph] .
  * Kaneta  _et al._ [2019] K. Kaneta, Y. Mambrini, and K. A. Olive, Phys. Rev. D 99, 063508 (2019), arXiv:1901.04449 [hep-ph] .
  * Mambrini and Olive [2021] Y. Mambrini and K. A. Olive, Phys. Rev. D 103, 115009 (2021), arXiv:2102.06214 [hep-ph] .
  * Hut [1977] P. Hut, Phys. Lett. B 69, 85 (1977).
  * Lee and Weinberg [1977] B. W. Lee and S. Weinberg, Phys. Rev. Lett. 39, 165 (1977).
  * Vysotsky  _et al._ [1977] M. I. Vysotsky, A. D. Dolgov, and Y. B. Zeldovich, JETP Lett. 26, 188 (1977).
  * ’t Hooft [1980] G. ’t Hooft, NATO Sci. Ser. B 59, 135 (1980).
  * Marrodán Undagoitia and Rauch [2016] T. Marrodán Undagoitia and L. Rauch, J. Phys. G 43, 013001 (2016), arXiv:1509.08767 [physics.ins-det] .
  * Rappoccio [2019] S. Rappoccio, Rev. Phys. 4, 100027 (2019), arXiv:1810.10579 [hep-ex] .
  * Gaskins [2016] J. M. Gaskins, Contemp. Phys. 57, 496 (2016), arXiv:1604.00014 [astro-ph.HE] .
  * Buttazzo  _et al._ [2013] D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice, F. Sala, A. Salvio, and A. Strumia, JHEP 12, 089 (2013), arXiv:1307.3536 [hep-ph] .
  * Alekhin  _et al._ [2012] S. Alekhin, A. Djouadi, and S. Moch, Phys. Lett. B 716, 214 (2012), arXiv:1207.0980 [hep-ph] .
  * Bednyakov  _et al._ [2015] A. V. Bednyakov, B. A. Kniehl, A. F. Pikelner, and O. L. Veretin, Phys. Rev. Lett. 115, 201802 (2015), arXiv:1507.08833 [hep-ph] .
  * Chung  _et al._ [1999] D. J. H. Chung, E. W. Kolb, and A. Riotto, Phys. Rev. D 60, 063504 (1999), arXiv:hep-ph/9809453 .
  * Kuzmin and Tkachev [1999b] V. A. Kuzmin and I. I. Tkachev, Phys. Rept. 320, 199 (1999b), arXiv:hep-ph/9903542 .
  * Aloisio  _et al._ [2015] R. Aloisio, S. Matarrese, and A. V. Olinto, JCAP 08, 024 (2015), arXiv:1504.01319 [astro-ph.HE] .
  * Abreu  _et al._ [2022a] P. Abreu _et al._ (Pierre Auger), (2022a), arXiv:2203.08854 [astro-ph.HE] .
  * McDonald [2002] J. McDonald, Phys. Rev. Lett. 88, 091304 (2002), arXiv:hep-ph/0106249 .
  * Hall  _et al._ [2010] L. J. Hall, K. Jedamzik, J. March-Russell, and S. M. West, JHEP 03, 080 (2010), arXiv:0911.1120 [hep-ph] .
  * Bernal  _et al._ [2017] N. Bernal, M. Heikinheimo, T. Tenkanen, K. Tuominen, and V. Vaskonen, Int. J. Mod. Phys. A 32, 1730023 (2017), arXiv:1706.07442 [hep-ph] .
  * Adam  _et al._ [2016] R. Adam _et al._ (Planck), Astron. Astrophys. 594, A1 (2016), arXiv:1502.01582 [astro-ph.CO] .
  * Andreassen  _et al._ [2018] A. Andreassen, W. Frost, and M. D. Schwartz, Phys. Rev. D 97, 056006 (2018), arXiv:1707.08124 [hep-ph] .
  * Markkanen  _et al._ [2018] T. Markkanen, S. Nurmi, A. Rajantie, and S. Stopyra, JHEP 06, 040 (2018), arXiv:1804.02020 [hep-ph] .
  * Aloisio  _et al._ [2004] R. Aloisio, V. Berezinsky, and M. Kachelriess, Phys. Rev. D 69, 094023 (2004), arXiv:hep-ph/0307279 .
  * Navarro  _et al._ [1996] J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J. 462, 563 (1996), arXiv:astro-ph/9508025 .
  * Einasto [1965] J. Einasto, Trudy Astrofizicheskogo Instituta Alma-Ata 5, 87 (1965).
  * Burkert [1995] A. Burkert, Astrophys. J. Lett. 447, L25 (1995), arXiv:astro-ph/9504041 .
  * Moore  _et al._ [1999] B. Moore, S. Ghigna, F. Governato, G. Lake, T. R. Quinn, J. Stadel, and P. Tozzi, Astrophys. J. Lett. 524, L19 (1999), arXiv:astro-ph/9907411 .
  * Ellis  _et al._ [2011] R. K. Ellis, W. J. Stirling, and B. R. Webber,  _QCD and collider physics_, Vol. 8 (Cambridge University Press, 2011).
  * Basu and Bhattacharjee [2004] R. Basu and P. Bhattacharjee, Phys. Rev. D 70, 023510 (2004), arXiv:hep-ph/0403299 .
  * Sarkar and Toldra [2002] S. Sarkar and R. Toldra, Nucl. Phys. B 621, 495 (2002), arXiv:hep-ph/0108098 .
  * Barbot and Drees [2003] C. Barbot and M. Drees, Astropart. Phys. 20, 5 (2003), arXiv:hep-ph/0211406 .
  * Kachelriess  _et al._ [2018] M. Kachelriess, O. E. Kalashev, and M. Y. Kuznetsov, Phys. Rev. D 98, 083016 (2018), arXiv:1805.04500 [astro-ph.HE] .
  * Alcantara  _et al._ [2019] E. Alcantara, L. A. Anchordoqui, and J. F. Soriano, Phys. Rev. D 99, 103016 (2019), arXiv:1903.05429 [hep-ph] .
  * Aab  _et al._ [2020a] A. Aab _et al._ (Pierre Auger), Phys. Rev. D 102, 062005 (2020a), arXiv:2008.06486 [astro-ph.HE] .
  * Abreu  _et al._ [2021] P. Abreu _et al._ (Pierre Auger), Eur. Phys. J. C 81, 966 (2021), arXiv:2109.13400 [astro-ph.HE] .
  * Abu-Zayyad  _et al._ [2013] T. Abu-Zayyad _et al._ (Telescope Array), Astrophys. J. Lett. 768, L1 (2013), arXiv:1205.5067 [astro-ph.HE] .
  * Aab  _et al._ [2020b] A. Aab _et al._ (Pierre Auger), JINST 15, P10021 (2020b), arXiv:2007.09035 [astro-ph.IM] .
  * Berezinsky and Mikhailov [1999] V. Berezinsky and A. A. Mikhailov, Phys. Lett. B 449, 237 (1999), arXiv:astro-ph/9810277 .
  * Benson  _et al._ [1999] A. Benson, A. W. Wolfendale, and A. Smialkowski, Astropart. Phys. 10, 313 (1999).
  * Medina-Tanco and Watson [1999] G. A. Medina-Tanco and A. A. Watson, Astropart. Phys. 12, 25 (1999), arXiv:astro-ph/9905240 .
  * Evans  _et al._ [2002] N. W. Evans, F. Ferrer, and S. Sarkar, Astropart. Phys. 17, 319 (2002), arXiv:astro-ph/0103085 .
  * Aloisio  _et al._ [2006] R. Aloisio, V. Berezinsky, and M. Kachelriess, Phys. Rev. D 74, 023516 (2006), arXiv:astro-ph/0604311 .
  * Siffert  _et al._ [2007] B. B. Siffert, B. Lazarotto, J. R. T. de Mello Neto, and A. Olinto, Braz. J. Phys. 37, 48 (2007).
  * Sommers [2001] P. Sommers, Astropart. Phys. 14, 271 (2001), arXiv:astro-ph/0004016 .
  * Abreu  _et al._ [2022b] P. Abreu _et al._ (Pierre Auger), (2022b).
  * Thielheim and Langhoff [1968] K. O. Thielheim and W. Langhoff, Journal of Physics A Mathematical General 1, 694 (1968).
  * Jansson and Farrar [2012] R. Jansson and G. R. Farrar, Astrophys. J. Lett. 761, L11 (2012), arXiv:1210.7820 [astro-ph.GA] .
  * Abreu  _et al._ [2022c] P. Abreu _et al._ (Pierre Auger), Astrophys. J. 933, 125 (2022c), arXiv:2205.14864 [astro-ph.HE] .
  * Savina [2021] P. Savina (Pierre Auger), Proc. 37th International Cosmic Ray Conference (Berlin, Germany) PoS (ICRC2021), 373 (2021).
  * Abreu  _et al._ [2022d] P. Abreu _et al._ (Pierre Auger), (2022d), arXiv:2209.05926 [astro-ph.HE] .
  * Anchordoqui  _et al._ [2021] L. A. Anchordoqui _et al._ , Astropart. Phys. 132, 102614 (2021), arXiv:2105.12895 [hep-ph] .
  * Capelle  _et al._ [1998] K. S. Capelle, J. W. Cronin, G. Parente, and E. Zas, Astropart. Phys. 8, 321 (1998), arXiv:astro-ph/9801313 .
  * Bertou  _et al._ [2002] X. Bertou, P. Billoir, O. Deligny, C. Lachaud, and A. Letessier-Selvon, Astropart. Phys. 17, 183 (2002), arXiv:astro-ph/0104452 .
  * Learned and Pakvasa [1995] J. G. Learned and S. Pakvasa, Astropart. Phys. 3, 267 (1995), arXiv:hep-ph/9405296 .
  * Athar  _et al._ [2000] H. Athar, M. Jezabek, and O. Yasuda, Phys. Rev. D 62, 103007 (2000), arXiv:hep-ph/0005104 .
  * Anchordoqui  _et al._ [2014] L. A. Anchordoqui _et al._ , JHEAp 1-2, 1 (2014), arXiv:1312.6587 [astro-ph.HE] .
  * Aab  _et al._ [2019] A. Aab _et al._ (Pierre Auger), JCAP 10, 022 (2019), arXiv:1906.07422 [astro-ph.HE] .
  * de Vega and Sanchez [2003] H. J. de Vega and N. G. Sanchez, Phys. Rev. D 67, 125019 (2003).
  * Kleiss  _et al._ [1986] R. Kleiss, W. J. Stirling, and S. D. Ellis, Comput. Phys. Commun. 40, 359 (1986).
  * Dudas  _et al._ [2020] E. Dudas, L. Heurtier, Y. Mambrini, K. A. Olive, and M. Pierre, Phys. Rev. D 101, 115029 (2020), arXiv:2003.02846 [hep-ph] .
  * Belavin  _et al._ [1975] A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Y. S. Tyupkin, Phys. Lett. B 59, 85 (1975).
  * Coleman [1979] S. R. Coleman, Subnucl. Ser. 15, 805 (1979).
  * Vainshtein  _et al._ [1982] A. I. Vainshtein, V. I. Zakharov, V. A. Novikov, and M. A. Shifman, Sov. Phys. Usp. 25, 195 (1982).
  * ’t Hooft [1976] G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976).
  * Kuzmin and Rubakov [1998] V. A. Kuzmin and V. A. Rubakov, Phys. Atom. Nucl. 61, 1028 (1998), arXiv:astro-ph/9709187 .
  * Zyla  _et al._ [2020] P. A. Zyla _et al._ (Particle Data Group), PTEP 2020 (2020), 10.1093/ptep/ptaa104, 083C01, https://academic.oup.com/ptep/article-pdf/2020/8/083C01/34673722/ptaa104.pdf .
  * Carone  _et al._ [2010] C. D. Carone, J. Erlich, and R. Primulando, Phys. Rev. D 82, 055028 (2010), arXiv:1008.0642 [hep-ph] .
  * Kolb and Turner [1990] E. W. Kolb and M. S. Turner,  _The Early Universe_, Vol. 69 (1990).
  * Giudice  _et al._ [2001] G. F. Giudice, E. W. Kolb, and A. Riotto, Phys. Rev. D 64, 023508 (2001), arXiv:hep-ph/0005123 .
  * Ade  _et al._ [2016] P. A. R. Ade _et al._ (Planck), Astron. Astrophys. 594, A20 (2016), arXiv:1502.02114 [astro-ph.CO] .
  * Garny  _et al._ [2018] M. Garny, A. Palessandro, M. Sandora, and M. S. Sloth, JCAP 02, 027 (2018), arXiv:1709.09688 [hep-ph] .
  * Abazajian  _et al._ [2016] K. N. Abazajian _et al._ (CMB-S4), (2016), arXiv:1610.02743 [astro-ph.CO] .
  * Ishino [2016] H. Ishino, Int. J. Mod. Phys. Conf. Ser. 43, 1660192 (2016).
  * Olinto  _et al._ [2021] A. V. Olinto _et al._ (POEMMA), JCAP 06, 007 (2021), arXiv:2012.07945 [astro-ph.IM] .
  * Hörandel [2021] J. R. Hörandel (GCOS), PoS 395, 027 (2021), arXiv:2203.01127 [astro-ph.HE] .
  * Espinosa  _et al._ [2008] J. R. Espinosa, G. F. Giudice, and A. Riotto, JCAP 05, 002 (2008), arXiv:0710.2484 [hep-ph] .
  * Calmet  _et al._ [2018] X. Calmet, I. Kuntz, and I. G. Moss, Found. Phys. 48, 110 (2018), arXiv:1701.02140 [hep-ph] .