# The CosmoVerse White Paper: Addressing observational tensions in cosmology with systematics and fundamental physics

Eleonora Di Valentino, Jackson Levi Said, Adam Riess, Agnieszka Pollo, Vivian Poulin, Adrià Gómez-Valent, Amanda Weltman, Antonella Palmese, Caroline D Huang, Carsten van de Bruck, et al.

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## ► To cite this version:

Eleonora Di Valentino, Jackson Levi Said, Adam Riess, Agnieszka Pollo, Vivian Poulin, et al. The CosmoVerse White Paper: Addressing observational tensions in cosmology with systematics and fundamental physics. Phys.Dark Univ., 2025, 49 (101965), pp.101965. 10.1016/j.dark.2025.101965 hal-05074695

HAL Id: hal-05074695

https://hal.science/hal-05074695v1

Submitted on 21 Aug 2025

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Review article

# The CosmoVerse White Paper: Addressing observational tensions in cosmology with systematics and fundamental physics

Eleonora Di Valentino<sup>1,†</sup>,*, Jackson Levi Said<sup>2,3</sup>, Adam Riess<sup>4,5</sup>, Agnieszka Pollo<sup>6</sup>, Vivian Poulin<sup>7</sup>, Adrià Gómez-Valent<sup>8</sup>, Amanda Weltman<sup>9</sup>, Antonella Palmese<sup>10</sup>, Caroline D. Huang<sup>11</sup>, Carsten van de Bruck<sup>12</sup>, Chandra Shekhar Saraf<sup>13</sup>, Cheng-Yu Kuo<sup>14</sup>, Cora Uhlemann<sup>15,16</sup>, Daniela Grandón<sup>17</sup>, Dante Paz<sup>18</sup>, Dominique Eckert<sup>19</sup>, Elsa M. Teixeira<sup>20</sup>, Emmanuel N. Saridakis<sup>21,22,23</sup>, Eoin Ó Colgáin<sup>24</sup>, Florian Beutler<sup>25</sup>, Florian Niedermann<sup>26</sup>, Francesco Bajardi<sup>27,28</sup>, Gabriela Barenboim<sup>29</sup>, Giulia Gubitosi<sup>30,28</sup>, Ilaria Musella<sup>31</sup>, Indranil Banik<sup>32</sup>, Istvan Szapudi<sup>33</sup>, Jack Singal<sup>34</sup>, Jaume Haro Cases<sup>35</sup>, Jens Chluba<sup>36</sup>, Jesús Torrado<sup>37</sup>, Jurgen Mifsud<sup>38,39</sup>, Karsten Jedamzik<sup>40</sup>, Khaled Said<sup>41</sup>, Konstantinos Dialektopoulos<sup>42,38</sup>, Laura Herold<sup>4</sup>, Leandros Perivolaropoulos<sup>43</sup>, Lei Zu<sup>6</sup>, Lluís Galbany<sup>44,45</sup>, Louise Breuval<sup>46</sup>, Luca Visinelli<sup>47,48</sup>, Luis A. Escamilla<sup>49</sup>, Luis A. Anchordoqui<sup>50</sup>, M.M. Sheikh-Jabbari<sup>51</sup>, Margherita Lembo<sup>52,53,54</sup>, Maria Giovanna Dainotti<sup>55,56</sup>, Maria Vincenzi<sup>57</sup>, Marika Asgari<sup>16</sup>, Martina Gerbino<sup>58</sup>, Matteo Forconi<sup>59</sup>, Michele Cantiello<sup>60</sup>, Michele Moresco<sup>61,62</sup>, Micol Benetti<sup>27,28</sup>, Nils Schöneberg<sup>63,64</sup>, Özgür Akarsu<sup>65</sup>, Rafael C. Nunes<sup>66,67</sup>, Reginald Christian Bernardo<sup>68</sup>, Ricardo Chávez<sup>69,70</sup>, Richard I. Anderson<sup>71</sup>, Richard Watkins<sup>72</sup>, Salvatore Capozziello<sup>30,27,28</sup>, Siyang Li<sup>4</sup>, Sunny Vagnozzi<sup>73,74</sup>, Supriya Pan<sup>75</sup>, Tommaso Treu<sup>76</sup>, Vid Irsic<sup>77,78</sup>, Will Handley<sup>79,80</sup>, William Giarè<sup>49</sup>, Yukei Murakami<sup>4</sup>, Abdolali Banihashemi<sup>81</sup>, Adèle Poudou<sup>82</sup>, Alan Heavens<sup>83</sup>, Alan Kogut<sup>84</sup>, Alba Domi<sup>85</sup>, Aleksander Łukasz Lenart<sup>86</sup>, Alessandro Melchiorri<sup>87</sup>, Alessandro Vadalà<sup>88,89,90</sup>, Alexandra Amon<sup>91,92</sup>, Alexander Bonilla Rivera<sup>93</sup>, Alexander Reeves<sup>94</sup>, Alexander Zhuk<sup>95,96,97</sup>, Alfio Bonanno<sup>98</sup>, Ali Övgün<sup>99</sup>, Alice Pisani<sup>100,101</sup>, Alireza Talebian<sup>102</sup>, Amare Abebe<sup>103</sup>, Amin Aboubrahim<sup>104</sup>, Ana Luisa González Morán<sup>105</sup>, András Kovács<sup>106,107</sup>, Andreas Lymeris<sup>108</sup>, Andreas Papatriantafyllou<sup>109</sup>, Andrew R. Liddle<sup>110</sup>, Andronikos Paliathanasis<sup>111,112,113</sup>, Andrzej Borowiec<sup>114</sup>, Anil Kumar Yadav<sup>115</sup>, Anita Yadav<sup>116</sup>, Anjan Ananda Sen<sup>117</sup>, Anjitha John William<sup>118</sup>, Anne Christine Davis<sup>119</sup>, Anowar J. Shajib<sup>120,121,122</sup>, Anthony Walters<sup>123,124,9</sup>, Anto Idicherian Lonappan<sup>125</sup>, Anton Chudaykin<sup>126</sup>, Antonio Capodagli<sup>127</sup>, Antonio da Silva<sup>110</sup>, Antonio De Felice<sup>128</sup>, Antonio Racioppi<sup>129</sup>, Araceli Soler Oficial<sup>130</sup>, Ariadna Montiel<sup>131</sup>, Arianna Favale<sup>132,8</sup>, Armando Bernui<sup>133</sup>, Arrianne Crystal Velasco<sup>134,135</sup>, Asta Heinesen<sup>136</sup>, Athanasios Bakopoulos<sup>137,109</sup>, Athanasios Chatzistavrakidis<sup>138</sup>, Bahman Khanpour<sup>139</sup>, Bangalore S. Sathyaprakash<sup>140</sup>, Bartek Zgirski<sup>141</sup>, Benjamin L'Huillier<sup>142</sup>, Benoit Famaey<sup>143</sup>, Bhuvnesh Jain<sup>144</sup>, Bing Zhang<sup>145</sup>, Biswajit Karmakar<sup>146</sup>, Branko Dragovich<sup>147</sup>, Brooks Thomas<sup>148</sup>, Carlos Correa<sup>149</sup>, Carlos G. Boiza<sup>150</sup>, Catarina Marques<sup>151,152</sup>, Celia Escamilla-Rivera<sup>153</sup>, Charalampos Tzerefos<sup>154,155</sup>, Chi Zhang<sup>156,157,158</sup>, Chiara De Leo<sup>159</sup>, Christian Pfeifer<sup>160</sup>,

<footnote>* Corresponding author.
E-mail addresses: e.divalentino@sheffield.ac.uk (E. Di Valentino), jackson.said@um.edu.mt (J.L. Said).</footnote>
Christine Lee¹⁰⁶, Christo Venter¹⁶¹, Cláudio Gomes¹⁶²,¹⁶³, Clecio Roque De bom¹⁶⁴, Cristian Moreno-Pulido¹⁶⁵, Damianos Iosifidis¹⁶⁶, Dan Grin¹⁶⁷, Daniel Blixt²⁷, Dan Scolnic¹⁶⁸, Daniele Oriti¹⁶⁹, Daria Dobrycheva¹⁷⁰, Dario Bettoni¹⁷¹,¹⁷², David Benisty¹⁷³, David Fernández-Arenas¹⁷⁴, David L. Wiltshire¹⁷⁵, David Sanchez Cid¹⁷⁶,¹⁷⁷, David Tamayo¹⁷⁸,¹⁵², David Valls-Gabaud¹⁷⁹, Davide Pedrotti⁷³, Deng Wang¹⁸⁰, Denitsa Staicova¹⁸¹, Despoina Totolou¹⁸², Diego Rubiera-Garcia¹⁸³, Dinko Milaković¹⁸⁴,¹⁸⁵, Dominic W. Pesce¹⁸⁶,¹⁸⁷, Dominique Sluse¹⁸⁸, Duško Borka¹⁸⁹, Ebrahim Yusofi¹⁹⁰,¹⁹¹, Elena Giusarma¹⁹², Elena Terlevich¹⁹³,⁸⁰, Elena Tomasetti¹⁹⁴,⁶², Elias C. Vagenas¹⁹⁵, Elisa Fazzari¹⁹⁶,¹⁹⁷,¹⁹⁸, Elisa G.M. Ferreira¹⁹⁹, Elvis Barakovic²⁰⁰, Emanuela Dimastrogiovanni²⁰¹, Emil Brinch Holm²⁰², Emil Mottola²⁰³, Emre Özülker²⁰⁴, Enrico Specogna²⁰⁴, Enzo Brocato²⁰⁵,²⁰⁶, Erik Jensko²⁰⁷, Erika Antonette Enriquez¹³⁴, Esha Bhatia²⁰⁸, Fabio Bresolin²⁰⁹, Felipe Avila¹³³, Filippo Bouchè²⁷,²¹⁰, Flavio Bombacigno²¹¹, Fotios K. Anagnostopoulos²¹², Francesco Pace²¹³,²¹⁴,²¹⁵, Francesco Sorrenti¹²⁶, Francisco S.N. Lobo¹¹⁰,²¹⁶, Frédéric Courbin²¹⁷,²¹⁸, Frode K. Hansen²¹⁹, Greg Sloan⁵,²²⁰, Gabriel Farrugia³⁸,²²¹, Gabriel Lynch²²², Gabriela Garcia-Arroyo²²³, Gabriella Raimondo⁶⁰, Gaetano Lambiase⁴⁷,⁴⁸, Gagandeep S. Anand⁵, Gaspard Poulot²⁰⁴, Genly Leon²²⁴,²²⁵, Gerasimos Kouniatalis²²⁶,¹⁵⁵, Germano Nardini²²⁷, Géza Csörnyei²²⁸, Giacomo Galloni²²⁹,²³⁰, Giada Bargiacchi²³¹, Giannis Papagiannopoulos²³², Giovanni Montani²³³,²³⁴, Giovanni Otalora²³⁵, Giulia De Somma³¹,²⁸, Giuliana Fiorentino²³⁶, Giuseppe Fanizza²³⁷, Giuseppe Gaetano Luciano²³⁸, Giuseppe Sarracino³¹, Gonzalo J. Olmo²¹¹,²³⁹, Goran S. Djordjević²⁴⁰, Guadalupe Cañas-Herrera²⁴¹, Hanyu Cheng²⁴²,²⁴³,²⁰⁴, Harry Desmond³², Hassan Abdalla¹¹², Houzun Chen²⁴⁴, Hsu-Wen Chiang²⁴⁵, Hume A. Feldman²⁴⁶, Hussain Gohar²⁴⁷, Ido Ben-Dayan²⁴⁸, Ignacio Sevilla-Noarbe¹⁷⁶, Ignatios Antoniadis²⁴⁹, Ilim Cimdiker²⁴⁷, Inês S. Albuquerque¹¹⁰, Ioannis D. Gialamas¹²⁹, Ippocratis Saltas²⁵⁰, Iryna Vavilova²⁵¹, Isidro Gómez-Vargas²⁵², Ismael Ayuso²⁵³, Ismailov Nariman Zeynalabdi²⁵⁴, Ivan De Martino¹⁷², Ivonne Zavala²⁵⁵, J. Alberto Vázquez¹⁵³, Jacobo Asorey²⁵⁶, Janusz Gluza²⁵⁷, Javier Rubio¹⁸³, Jenny G. Sorce²⁵⁸,²⁵⁹, Jenny Wagner²⁶⁰,²⁶¹,²⁶², Jeremy Sakstein²⁶³, Jessica Santiago²⁶⁴, Jim Braatz²⁶⁵, Joan Solà Peracaula⁸, John Blakeslee²⁶⁶, John Webb⁸⁰, Jose A.R. Cembranos²⁶⁷, José Pedro Mimoso¹¹⁰, Joseph Jensen²⁶⁸, Juan García-Bellido²⁶⁹, Judit Prat²⁷⁰, Kathleen Sammut³⁸, Kay Lehnert²⁷¹, Keith R. Dienes²⁷²,²⁷³, Kishan Deka⁶, Konrad Kuijken²⁷⁴, Krishna Naidoo²⁷⁵, László Árpád Gergely²⁷⁶,²⁷⁷, Laur Järv²⁷⁸, Laura Mersini-Houghton²⁷⁹, Leila L. Graef²⁸⁰, Léo Vacher²⁸¹, Levon Pogosian²⁸², Lilia Anguelova²⁸³, Lindita Hamolli²⁸⁴, Lu Yin²⁸⁵,²⁸⁶, Luca Caloni²²⁹,²³⁰,²⁸⁷, Luca Izzo²⁸⁸, Lucas Macri²⁸⁹, Luis E. Padilla²⁹⁰, Luz Ángela García²⁹¹, Maciej Bilicki²⁹², Mahdi Najafi¹⁹⁶,²⁹³, Manolis Plionis²¹,²⁹⁴,²⁹⁵, Manuel Gonzalez-Espinoza²⁹⁶,²⁹⁷, Manuel Hohmann²⁷⁸, Marcel A. van der Westhuizen¹⁰³, Marcella Marconi³¹, Marcin Postolak²⁹⁸, Marco de Cesare²⁷,²⁸, Marco Regis²⁹⁹, Marek Biesiada⁶, Maret Einasto³⁰⁰, Margus Saal³⁰¹, Maria Caruana³⁸, Maria Petronikolou³⁰²,¹⁵⁵, Mariam Bouhmadi-López³⁰³,³⁰⁴, Mariana Melo¹⁵¹,¹⁵², Mariaveronica De Angelis²⁰⁴, Marie-Noëlle Célérier³⁰⁵, Marina Cortês¹¹⁰, Mark Reid¹⁸⁶, Markus Michael Rau¹⁶,³⁰⁶, Martin S. Sloth³⁰⁷, Martti Raidal¹²⁹, Masahiro Takada³⁰⁸, Masoume Reyhani³⁰⁹,²⁹³, Massimiliano Romanello³¹⁰,³¹¹, Massimo Marengo¹²⁷, Mathias Garny³¹², Matías Leizerovich³¹³,³¹⁴, Matteo Martinelli³¹⁵,³¹⁶, Matteo Tagliazucchi³¹⁷,⁶², Mehmet Demirci³¹⁸, Miguel A.S. Pinto¹¹⁰,²¹⁶, Miguel A. Sabogal⁶⁶, Miguel A. García-Aspeitia³¹⁹, Milan Milošević³²⁰, Mina Ghodsi³²¹,¹⁰⁶, Mustapha Ishak³²², Nelson J. Nunes¹¹⁰, Nick Samaras³²³, Nico Hamaus³²⁴, Nico Schuster¹⁰⁰,³²⁴, Nicola Borghi³¹⁷,⁶²,³²⁵, Nicola Deiosso¹⁷⁶, Nicola Tamanini³²⁶, Nicolao Fornengo³²⁷, Nihan Katirci³²⁸, Nikolaos E. Mavromatos¹⁰⁹,²⁹², Nicholas Petropoulos³³⁰, Nikolina Šarčević³³¹, Nils A. Nilsson³³²,³³³, Nima Khosravi³³⁴,³³⁵, Noemi Frusciante³⁰, Octavian Postavaru³³⁶, Oem Trivedi³³⁷,³³⁸, Oleksii Sokoliuk³³⁹,³⁴⁰,³⁴¹, Olga Mena¹⁸⁰, Paloma Morilla³⁴², Paolo Campeti³⁴³,³⁴⁴, Paolo Salucci²⁸¹, Paula Boubel³⁴⁵, Paweł Bielewicz⁶, Pekka Heinämäki³⁴⁶, Petar Suman³⁴⁷, Petros Asimakis³⁴⁸, Pierros Ntelis¹⁰⁰, Pran Nath³⁴⁹, Predrag Jovanović³⁵⁰, Purba Mukherjee¹¹⁷,³⁵¹, Radosław Wojtak³⁵², Rafaela Gsponer³⁵³, Rafid H. Dejrah³⁵⁴,
Rahul Shah 351, Rasmi Hajjar 180, Rebecca Briffa 38, Rebecca Habas 355, Reggie C. Pantig 356, Renier Mendoza 134,135, Riccardo Della Monica 172, Richard Stiskalek 357, Rishav Roshan 358, Rita B. Neves 49, Roberto Molinaro 31, Roberto Terlevich 359,80,360, Rocco D'Agostino 88,361, Rodrigo Sandoval-Orozco 362, Ronaldo C. Batista 363, Ruchika 172, Ruth Lazkoz 364, Saeed Rastgoo 365,366,367, Sahar Mohammadi 368, Salvatore Samuele Sirletti 369,370,371, Sandeep Haridasu 372,373, Sanjay Mandal 374, Saurya Das 375, Sebastian Bahamonde 376,377, Sebastian Grandis 378, Sebastian Trojanowski 6, Sergei D. Odintsov 379,380, Sergij Mazurenko 381, Shahab Joudaki 176,382, Sherry H. Suyu 383,384, Shouvik Roy Choudhury 385, Shruti Bhatporia 9, Shun-Sheng Li 386,387, Simeon Bird 388, Simon Birrer 389, Simone Paradiso 390, Simony Santos da Costa 391,392, Sofia Contarini 393, Sophie Henrot-Versillé 394, Spyros Basilakos 21,395,396, Stefano Casertano 5, Stefano Gariazzo 397, Stylianos A. Tsilioukas 398,21, Surajit Kalita 399,9, Suresh Kumar 400, Susana J. Landau 401, Sveva Castello 126, Swayamtrupta Panda 402,403, Tanja Petrushevska 404, Thanasis Karakasis 109, Thejs Brinckmann 405,406, Tiago B. Gonçalves 110,216, Tiziano Schiavone 407,408, Tom Abel 409, Tomi Koivisto 166,410, Torsten Bringmann 411, Umut Demirbozan 412, Utkarsh Kumar 413,414, Valerio Marra 415,184,185, Maurice H.P.M. van Putten 416,417, Vasileios Kalaitzidis 418, Vasiliki A. Mitsou 419, Vasilios Zarikas 420, Vedad Pasic 421, Venus Keus 422,423, Verónica Motta 424, Vesna Borka Jovanović 189, Víctor H. Cárdenas 424, Vincenzo Ripepi 31, Vincenzo Salzano 247, Violetta Impellizzeri 425, Vitor da Fonseca 110, Vittorio Ghirardini 426,427, Vladas Vansevičius 428, Weiqiang Yang 429, Wojciech Hellwing 292, Xin Ren 430,431, Yu-Min Hu 430,432, Yuejia Zhai 49, Abdul Malik Sultan 433, Abdurakhmon Nosirov 434, Adrienn Pataki 435, Alessandro Santoni 436,437, Aliya Batool 433, Amlan Chakraborty 438, Aneta Wojnar 439,440, Arman Tursunov 441,442, Avik De 443, Ayush Hazarika 444, Baojiu Li 445, Benjamin Bose 446,447, Bivudutta Mishra 448, Bobomurat Ahmedov 449,450, Claudia Scócola 451, Crescenzo Tortora 31, D'Arcy Kenworthy 452, Daniel E. Holz 453, David F. Mota 454, David S. Pereira 110, Devon M. Williams 455, Dillon Brout 456, Dong Ha Lee 49, Eduardo Guendelman 457, Edward Olex 458, Emanuelly Silva 459, Emre Onur Kahya 460, Eva-Maria Mueller 461, Felipe Andrade-Oliveira 462, Feven Markos Hunde 292, F.R. Joaquim 463,464, Florian Pacaud 465, Francis-Yan Cyr-Racine 466, F. Pozo Nuñez 467, Gábor Rácz 468, Gene Carlo Belinario 469, Geraint F. Lewis 470, Gergely Dálya 471,472, Giorgio Laverda 473, Guido Risaliti 474,475, Guillermo Franco-Abellán 476, Hayden Zammit 38, Hayley Camilleri 38, Helene M. Courtois 477, Hooman Moradpour 478, Igor de Oliveira Cardoso Pedreira 479, Ilídio Lopes 480, István Csabai 481, James W. Rohlf 482, Jana Bogdanoska 483, Javier de Cruz Pérez 484, Joan Bachs-Esteban 473, Joseph Sultana 485, Julien Lesgourgues 486, Jun-Qian Jiang 487, Karem Peñaló Castillo 50, Kimet Jusufi 488, Lavinia Heisenberg 489, Laxmipriya Pati 166, Lón V.E. Koopmans 490, Lokesh kumar Duchaniya 491, Lucas Lombriser 492, María Pérez Garrote 172, Mariano Domínguez 493, Marine Samsonyan 494, Mark Pace 495,38, Martin Krššák 496, Masroor C. Pookkillath 497, Matteo Peronaci 132, Matteo Piani 473, Matthildi Raftogianni 137, Meet J. Vyas 338, Melina Michalopoulou 137, Merab Gogberashvili 498, Michael Klasen 499, Michele Cicoli 500,501, Miguel Quartin 502,503, Miguel Zumalacárregui 504, Milan S. Dimitrijević 505, Milos Dordevic 506, Mindaugas Karčiauskas 507, Morgan Le Delliou 508,509,510, Nastassia Grimm 126, Nicolás Augusto Kozameh 493, Nicoleta Voicu 42, Nicolina Pop 511, Nikos Chatzifotis 109, Odil Yunusov 512, Oliver Fabio Piattella 513,514,515, Pedro da Silveira Ferreira 516, Péter Raffai 435,517, Peter Schupp 518, Pilar Ruiz-Lapuente 519, Pradyumn Kumar Sahoo 491, Roberto V. Maluf 520, Ruth Durrer 521, S.A. Kadam 522, Sabino Matarrese 523, Samuel Brieden 524, Santiago González-Gaitán 525, Santosh V. Lohakare 491, Scott Watson 526,527, Shao-Jiang Wang 528,529, Simão Marques Nunes 110, Soumya Chakrabarti 530, Subinoy Das 438, Suvodip Mukherjee 531, Tajron Jurić 532, Tessa Baker 533, Theodoros Nakas 332, Tiago Barreiro 110,534, Upala Mukhopadhyay 535, Veljko Vujčić 536, Violetta Sagun 537, Vladimir A. Srećković 538, Wangzheng Zhang 539, Yo Toda 540, Yun-Song Piao 487,541,542, Zahra Davari 543, for The CosmoVerse Network

<footnote n="1">¹ School of Mathematical and Physical Sciences, University of Sheffield, Hounsfield Road, Sheffield, S3 7RH, United Kingdom</footnote>
<footnote n="2">² Institute of Space Sciences and Astronomy, University of Malta, Malta</footnote>

<footnote n="3">³ Department of Physics, University of Malta, Malta</footnote>

<footnote n="4">⁴ Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD, 21218, USA</footnote>

<footnote n="5">⁵ Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA</footnote>

<footnote n="6">⁶ National Centre for Nuclear Research, Pasteura 7, Warszawa, 02-093, Poland</footnote>

<footnote n="7">⁷ Laboratoire Univers & Particules de Montpellier (LUPM), CNRS & Université de Montpellier (UMR-5299), Place Eugène Bataillon, F-34095 Cedex 05, Montpellier, France</footnote>

<footnote n="8">⁸ Departament de Física Quàntica i Astrofísica and Institut de Ciències del Cosmos, Universitat de Barcelona, Av. Diagonal 647, Barcelona, E-08020, Spain</footnote>

<footnote n="9">⁹ High Energy Physics, Cosmology and Astrophysics Theory (HEPCAT) Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Cape Town, 7700, South Africa</footnote>

<footnote n="10">¹⁰ McWilliams Center for Cosmology and Astrophysics, Department of Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA, 15213, USA</footnote>

<footnote n="11">¹¹ Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA, 02138, USA</footnote>

<footnote n="12">¹² School of Mathematical and Physical Sciences, The University of Sheffield, Hounsfeld Road, Sheffield, S3 7RH, United Kingdom</footnote>

<footnote n="13">¹³ Korea Astronomy and Space Science Institute, 776 Daedeo-daero, Yuseong-gu, Daejeon, Republic of Korea</footnote>

<footnote n="14">¹⁴ Physics Department, National Sun Yat-Sen University, No. 70, Lien-Hai Rd, Kaosiung City, 80424, Taiwan, ROC</footnote>

<footnote n="15">¹⁵ Fakultät für Physik, Universität Bielefeld, Postfach 100131, Bielefeld, 33501, Germany</footnote>

<footnote n="16">¹⁶ School of Mathematics, Statistics and Physics, Newcastle University, Herschel Building, Newcastle-upon-Tyne, NE1 7RU, UK</footnote>

<footnote n="17">¹⁷ Mathematical Institute, Leiden University, Gorlaeus Gebouw, Einsteinweg 55, Leiden, NL-2333 CC, The Netherlands</footnote>

<footnote n="18">¹⁸ Instituto de Astronomía Teórica y Experimental, UNC-Conicet, Laprida 854, Ciudad de Córdoba, Argentina</footnote>

<footnote n="19">¹⁹ Department of Astronomy, University of Geneva, Ch. d'Ecogia 16, Versoix, CH-1290, Switzerland</footnote>

<footnote n="20">²⁰ Laboratoire Univers & Particules de Montpellier, CNRS & Université de Montpellier (UMR-5299), Montpellier, 34095, France</footnote>

<footnote n="21">²¹ National Observatory of Athens, Lofos Nymfon, Athens, 11852, Greece</footnote>

<footnote n="22">²² CAS Key Laboratory for Researches in Galaxies and Cosmology, School of Astronomy and Space Science, University of Science and Technology of China, Hefei, 230026, China</footnote>

<footnote n="23">²³ Departamento de Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280, Antofagasta, Chile</footnote>

<footnote n="24">²⁴ Atlantic Technological University, Ash Lane, Sligo, Ireland</footnote>

<footnote n="25">²⁵ Institute for Astronomy, University of Edinburgh Royal Observatory Edinburgh, Blackford Hill, Edinburgh, EH9 3HJ, UK</footnote>

<footnote n="26">²⁶ Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes Alfvéns väg 12, Stockholm, SE-106 91, Sweden</footnote>

<footnote n="27">²⁷ Scuola Superiore Meridionale, Via Mezzocannone 4, Napoli, 80134, Italy</footnote>

<footnote n="28">²⁸ Istituto Nazionale di Fisica Nucleare, Fizione di Napoli, Complesso Univ. Monte S. Angelo, Napoli, I-80126, Italy</footnote>

<footnote n="29">²⁹ Departament de Física Teòrica and IFIC, Universitat de València-CSIC, Burjassot, E-46100, Spain</footnote>

<footnote n="30">³⁰ Dipartimento di Fisica Ettore Pancini, Università di Napoli Federico II, Complesso Univ. Monte S. Angelo, Napoli, I-80126, Italy</footnote>

<footnote n="31">³¹ INAF-Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, Napoli, 80131, Italy</footnote>

<footnote n="32">³² Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama Building, Burnaby Road, Portsmouth, PO1 3FX, UK</footnote>

<footnote n="33">³³ Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI, 96822, USA</footnote>

<footnote n="34">³⁴ Physics Department, University of Richmond, Richmond, VA, 23173, USA</footnote>

<footnote n="35">³⁵ Department of Mathematics, Universitat Politécnica de Catalunya, Carrer Colom 15, Terrassa, 08222, Spain</footnote>

<footnote n="36">³⁶ Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, Manchester, M13 9PL, UK</footnote>

<footnote n="37">³⁷ Instituto de Estructura de la Materia, CSIC, Serrano 121, Madrid, 28006, Spain</footnote>

<footnote n="38">³⁸ Institute of Space Sciences and Astronomy, University of Malta, Malta</footnote>

<footnote n="39">³⁹ Department of Physics, University of Malta, Malta</footnote>

<footnote n="40">⁴⁰ University of Montpellier, France</footnote>

<footnote n="41">⁴¹ School of Mathematics and Physics, University of Queensland, 4072, Australia</footnote>

<footnote n="42">⁴² Department of Mathematics and Computer Science, Transilvania University of Brasov, Romania</footnote>

<footnote n="43">⁴³ Department of Physics, University of Ioannina, Ioannina, 45110, Greece</footnote>

<footnote n="44">⁴⁴ Institute of Space Sciences (ICE-CSIC), Campus UAB, Carrer de Can Magrans, s/n, Barcelona, E-08193, Spain</footnote>

<footnote n="45">⁴⁵ Institut d'Estudis Espacials de Catalunya (IEEC), Castelldefels, Barcelona, 08860, Spain</footnote>

<footnote n="46">⁴⁶ European Space Agency (ESA), ESA Office, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA</footnote>

<footnote n="47">⁴⁷ Dipartimento di Fisica "E.R. Caianiello", Università degli Studi di Salerno, Via G. Paolo II, Fisciano, SA, 84084, Italy</footnote>

<footnote n="48">⁴⁸ INFN - Gruppo Collegato di Salerno, Via G.Paolo II, Fisciano, SA, 84084, Italy</footnote>

<footnote n="49">⁴⁹ School of Mathematical and Physical Sciences, University of Sheffield, Hounsfeld Road, Sheffield, S3 7RH, United Kingdom</footnote>

<footnote n="50">⁵⁰ Department of Physics and Astronomy, Lehman College, City University of New York, New York, NY, 10468, USA</footnote>

<footnote n="51">⁵¹ School of Physics, Inst. for Research in Fundamental Science (IPM), P.O.Box: 19395-5531, Tehran, Iran</footnote>

<footnote n="52">⁵² Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, via Saragat 1, Ferrara, I-44122, Italy</footnote>

<footnote n="53">⁵³ Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via G. Saragat 1, Ferrara, I-44122, Italy</footnote>

<footnote n="54">⁵⁴ Sorbonne Université, CNRS, UMR 7095, Institut d'Astrophysique de Paris, 98 bis bd Arago, Paris, 75014, France</footnote>

<footnote n="55">⁵⁵ National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan</footnote>

<footnote n="56">⁵⁶ Japan and Nevada Center for Astrophysics, University of Nevada, Las Vegas, 4505 Maryland Parkway, Las Vegas, 89154, USA</footnote>

<footnote n="57">⁵⁷ Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, United Kingdom</footnote>

<footnote n="58">⁵⁸ Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe Saragat, 1, Ferrara, 44122, Italy</footnote>

<footnote n="59">⁵⁹ Physics Department and INFN sezione di Ferrara, Università degli Studi di Ferrara, via Saragat 1, Ferrara, I-44122, Italy</footnote>

<footnote n="60">⁶⁰ INAF, Osservatorio Astronomico d'Abruzzo, Via Maggini snc, Teramo, I-64100, Italy</footnote>

<footnote n="61">⁶¹ Università di Bologna, Dipartimento di Fisica e Astronomia "Augusto Righi", via Piero Gobetti 93/2, Bologna, I-40129, Italy</footnote>

<footnote n="62">⁶² INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Piero Gobetti 93/3, Bologna, I-40129, Italy</footnote>

<footnote n="63">⁶³ University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstrasse 1, Munich, 81677, Germany</footnote>

<footnote n="64">⁶⁴ Excellence Cluster ORIGINS, Boltzmannstr 2, Garching, 85748, Germany</footnote>

<footnote n="65">⁶⁵ Department of Physics, Istanbul Technical University, Maslak, Istanbul, 34469, Türkiye</footnote>

<footnote n="66">⁶⁶ Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, 91501-970, Brazil</footnote>

<footnote n="67">⁶⁷ Divisão de Astrofísica, Instituto Nacional de Pesquisas Espaciais, Avenida dos Astronautas 1758, São José dos Campos, São Paulo, 12227-010, Brazil</footnote>

<footnote n="68">⁶⁸ Asia Pacific Center for Theoretical Physics, Pohang, 37673, Korea</footnote>

<footnote n="69">⁶⁹ Universidad Nacional Autónoma de México, Instituto de Radioastronomía y Astrofísica, Morelia, Michoacán, 58090, México</footnote>

<footnote n="70">⁷⁰ Secretaría de Ciencia, Humanidades, Tecnología e Innovación, Av. Insurgentes Sur 1582, Ciudad de México, 03940, México</footnote>

<footnote n="71">⁷¹ Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, Versoix, 1290, Switzerland</footnote>

<footnote n="72">⁷² Department of Physics, Willamette University, Salem, OR, 97301, USA</footnote>

<footnote n="73">⁷³ Department of Physics, University of Trento, Via Sommarive 14, Povo, TN, 38123, Italy</footnote>

<footnote n="74">⁷⁴ Trento Institute for Fundamental Physics and Applications-INFN, Via Sommarive 14, Povo, TN, 38123, Italy</footnote>

<footnote n="75">⁷⁵ Department of Mathematics, Presidency University, 86/1 College Street, Kolkata, 700073, India</footnote>

<footnote n="76">⁷⁶ Department of Physics and Astronomy University of California, USA</footnote>

<footnote n="77">⁷⁷ Kavli Institute for Cosmology (KICC), Madingley Road, Cambridge, CB3 0HA, United Kingdom</footnote>
<footnote n="78">⁷⁸ Centre for Astrophysics Research, Department of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hatfield, AL10 9AB, United Kingdom</footnote>

<footnote n="79">⁷⁹ Kavli Institute for Cosmology (KICC), Madingley Road, Cambridge, CB3 0HA, UK</footnote>

<footnote n="80">⁸⁰ Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 9AL, UK</footnote>

<footnote n="81">⁸¹ Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, Oslo, 0315, Norway</footnote>

<footnote n="82">⁸² Laboratoire Univers et Particules de Montpellier (LUPM), CNRS et Université de Montpellier (UMR-5299), Place Eugène Bataillon, F-34095 Cedex 05, Montpellier, France</footnote>

<footnote n="83">⁸³ Imperial Centre for Inference and Cosmology (ICIC), Department of Physics, Blackett Laboratory, Imperial College London, Prince Consort Road, London, SW7 2AZ, UK</footnote>

<footnote n="84">⁸⁴ NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD, 20771, USA</footnote>

<footnote n="85">⁸⁵ Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Department of Physics, Nikolaus-Fiebiger-Straße 2, Erlangen, 91058, Germany</footnote>

<footnote n="86">⁸⁶ Astronomical Observatory, Jagiellonian University in Kraków, ul. Orla 171, Kraków, 30-244, Poland</footnote>

<footnote n="87">⁸⁷ Dipartimento di Fisica “G. Marconi”, Università di Roma Sapienza, P.le Aldo Moro 2, Rome, 00185, Italy</footnote>

<footnote n="88">⁸⁸ INAF - Osservatorio Astronomico di Roma, Via Frascati 33, Monte Porzio Catone, 00078, Italy</footnote>

<footnote n="89">⁸⁹ INFN - Sezione di Roma, Piazzale Aldo Moro, 2 - c/o Dipartimento di Fisica, Roma, 00185, Italy</footnote>

<footnote n="90">⁹⁰ Dipartimento di Fisica, Università degli Studi di Roma Tor Vergata, via della Ricerca Scientifica 1, Rome, I-00133, Italy</footnote>

<footnote n="91">⁹¹ Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ, 08544, USA</footnote>

<footnote n="92">⁹² Kavli Institute for Cosmology (KICC), University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK</footnote>

<footnote n="93">⁹³ Instituto de Física, Universidade Federal Fluminense, Niterói, RJ, 24210-346, Brazil</footnote>

<footnote n="94">⁹⁴ Institute for Particle Physics and Astrophysics, ETH Zürich, Wolfgang-Pauli-Strasse 27, Zürich, CH8093, Switzerland</footnote>

<footnote n="95">⁹⁵ Astronomical Observatory, Odessa I.I. Mechnikov National University, Dvoryanskaya St. 2, Odessa, 65082, Ukraine</footnote>

<footnote n="96">⁹⁶ Center for Advanced Systems Understanding, Untermarkt 20, Görlitz, 02826, Germany</footnote>

<footnote n="97">⁹⁷ Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, Dresden, 01328, Germany</footnote>

<footnote n="98">⁹⁸ INAF - Osservatorio Astrofísico di Catania, Via S.Sofia 78, 1-95123 Catania, Italy, INFN, Sezione di Catania, Via S.Sofia 64, Catania, I-95123, Italy</footnote>

<footnote n="99">⁹⁹ Physics Department, Eastern Mediterranean University, Famagusta, Cyprus</footnote>

<footnote n="100">¹⁰⁰ Aix-Marseille Université, CNRS/IN2P3, CPPM, Marseille, France</footnote>

<footnote n="101">¹⁰¹ Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ, 08544, USA</footnote>

<footnote n="102">¹⁰² School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran</footnote>

<footnote n="103">¹⁰³ Centre for Space Research, North-West University, Potchefstroom, South Africa</footnote>

<footnote n="104">¹⁰⁴ Department of Physics and Astronomy, Union College, 807 Union Street, Schenectady, NY, 12308, USA</footnote>

<footnote n="105">¹⁰⁵ Instituto Nacional de Astrofísica, Óptica y Electrónica, Puebla, México</footnote>

<footnote n="106">¹⁰⁶ MTA-CSFK Lendület Large-scale Structure Research Group, Konkoly Thege Miklós út 15-17, Budapest, H-1121, Hungary</footnote>

<footnote n="107">¹⁰⁷ Konkoly Observatory, HUN-REN CSFK, MTA Centre of Excellence, Konkoly Thege Miklós út 15-17., Budapest, H-1121, Hungary</footnote>

<footnote n="108">¹⁰⁸ Department of Physics, University of Patras, Patras, 26500, Greece</footnote>

<footnote n="109">¹⁰⁹ National Technical University of Athens, School of Applied Mathematics and Physical Sciences, Physics Division, Athens, GR15780, Greece</footnote>

<footnote n="110">¹¹⁰ Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, Lisbon, P-1749-016, Portugal</footnote>

<footnote n="111">¹¹¹ School for Data Science and Computational Thinking and Department of Mathematical Sciences, Stellenbosch University, Stellenbosch, 7602, South Africa</footnote>

<footnote n="112">¹¹² Centre for Space Research, North-West University, Potchefstroom, 2520, South Africa</footnote>

<footnote n="113">¹¹³ Departamento de Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280, Antofagasta, Chile</footnote>

<footnote n="114">¹¹⁴ Institute of Theoretical Physics, University of Wroclaw, pl. M. Borna 9, Wroclaw, 50-204, Poland</footnote>

<footnote n="115">¹¹⁵ Department of Physics, United College of Engineering and Research, Greater Noida, 201310, India</footnote>

<footnote n="116">¹¹⁶ Department of Mathematics, Indira Gandhi University, Meerpur, Haryana, 122502, India</footnote>

<footnote n="117">¹¹⁷ Centre For Theoretical Physics, Jamia Millia Islamia, New Delhi, India</footnote>

<footnote n="118">¹¹⁸ Center for Theoretical Physics, Polish Academy of Sciences, al. Lotników 32/46, Warsaw, 02-668, Poland</footnote>

<footnote n="119">¹¹⁹ DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WA, UK</footnote>

<footnote n="120">¹²⁰ Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL, 60637, USA</footnote>

<footnote n="121">¹²¹ Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, 60637, USA</footnote>

<footnote n="122">¹²² Center for Astronomy, Space Science and Astrophysics, Independent University, Bangladesh, Dhaka, 1229, Bangladesh</footnote>

<footnote n="123">¹²³ Astrophysics Research Centre, University of KwaZulu-Natal, Westville Campus, Durban, 4041, South Africa</footnote>

<footnote n="124">¹²⁴ School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Westville Campus, Durban, 4041, South Africa</footnote>

<footnote n="125">¹²⁵ Department of Physics, 9500 Gilman Drive, San Diego, CA, 92122, USA</footnote>

<footnote n="126">¹²⁶ Département de Physique Théorique and Center for Astroparticle Physics, Université de Genève, 24 quai Ernest Ansermet, Genève, 1211 4, Switzerland</footnote>

<footnote n="127">¹²⁷ Florida State University, Department of Physics, 77 Chieftain Way, Tallahassee, FL, 32306, USA</footnote>

<footnote n="128">¹²⁸ Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan</footnote>

<footnote n="129">¹²⁹ National Institute of Chemical Physics and Biophysics, Rávala 10, Tallinn, 10143, Estonia</footnote>

<footnote n="130">¹³⁰ Department of Physics and EHU Quantum Center, University of the Basque Country UPV/EHU, Barrio Sarriena s/n, Leioa, 48940, Spain</footnote>

<footnote n="131">¹³¹ Physics Department, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (Cinvestav), PO. Box 14-740, Av. Instituto Politécnico Nacional 2508, Mexico City, Mexico</footnote>

<footnote n="132">¹³² Dipartimento di Fisica and INFN Sezione di Roma 2, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, Rome, 00133, Italy</footnote>

<footnote n="133">¹³³ Observatório Nacional, Rua General José Cristino 77, São Cristóvão, Rio de Janeiro, RJ, 20921-400, Brasil</footnote>

<footnote n="134">¹³⁴ Institute of Mathematics, University of the Philippines Diliman, Quezon City, Metro Manila, Philippines</footnote>

<footnote n="135">¹³⁵ Computational Science Research Center, University of the Philippines Diliman, Quezon City, Metro Manila, Philippines</footnote>

<footnote n="136">¹³⁶ Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DK-2100, Denmark</footnote>

<footnote n="137">¹³⁷ Division of Applied Analysis, Department of Mathematics, University of Patras, Rio Patras, GR-26504, Greece</footnote>

<footnote n="138">¹³⁸ Division of Theoretical Physics, Rudjer Bošković Institute, Bijenička 54, Zagreb, 10000, Croatia</footnote>

<footnote n="139">¹³⁹ Department of Electrical Engineering, Mazandaran University of Science and Technology, Babol, Iran</footnote>

<footnote n="140">¹⁴⁰ Institute for Gravitation and the Cosmos, Department of Physics and Department of Astronomy and Astrophysics, Pennsylvania State University, University Park, Pennsylvania, 16802, USA</footnote>

<footnote n="141">¹⁴¹ Universidad de Concepción, Departamento de Astronomía, Casilla 160-C, Concepción, Chile</footnote>

<footnote n="142">¹⁴² Department of Physics and Astronomy, Sejong University, Seoul, 05006, Korea</footnote>

<footnote n="143">¹⁴³ Université de Strasbourg, CNRS UMR 7550, Observatoire astronomique de Strasbourg, 11 rue de l'Université, Strasbourg, 67000, France</footnote>

<footnote n="144">¹⁴⁴ Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, 19104, USA</footnote>

<footnote n="145">¹⁴⁵ Nevada Center for Astrophysics and Department of Physics and Astronomy, University of Nevada, Las Vegas, Las Vegas, NV, 89154, USA</footnote>

<footnote n="146">¹⁴⁶ Institute of Physics, University of Silesia in Katowice, Poland</footnote>

<footnote n="147">¹⁴⁷ Mathematical Institute of the Serbian Academy of Sciences and Arts, Kneza Mihaila 36, Belgrade, 11000, Serbia</footnote>

<footnote n="148">¹⁴⁸ Department of Physics, Lafayette College, Easton, PA, 18042, USA</footnote>

<footnote n="149">¹⁴⁹ Max Planck Institute for Extraterrestrial Physics, Garching, Germany</footnote>

<footnote n="150">¹⁵⁰ Department of Physics, University of the Basque Country UPV/EHU, P.O. Box 644, Bilbao, 48080, Spain</footnote>
151 Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, Porto, 4150-007, Portugal

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

153 Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Cuernavaca, México

154 Department of Physics, National & Kapodistrian University of Athens, Zografou Campus, Athens, GR 157 73, Greece

155 National Observatory of Athens, Lofos Nymfon, Athens, 11852, Greece

156 Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chines Academy of Sciences, Nanjing, 210023, People's Republic of China

157 School of Astronomy and Space Science, University of Science and Technology of China, Hefei, 230026, People's Republic of China

158 SISSA - International School for Advanced Studies, Via Bonomea 265, Trieste, 34136, Italy

159 Sapienza University, Piazzale Aldo Moro, c/o Dipartimento di Fisica, Edificio E. Fermi, Roma, Italy

160 Center of applied space technology and microgravity (ZARM), University of Bremen, Am Fallturm 2, Bremen, 28359, Germany

161 Centre for Space Research, North-West University, Private Bag X6001, Potchefstroom, South Africa

162 Centro de Física das Universidades do Minho e do Porto, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n, Porto, 4169-007, Portugal

163 Universidade dos Açores Instituto de Investigação em Ciências do Mar - OKEANOS, Campus da Horta, Rua Professor Doutor Frederico Machado 4, Horta, 9900-140, Portugal

164 Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Rio de Janeiro, RJ, CEP 22290-180, Brazil

165 Departamento d'Informática, Matemática Aplicada i Estadística, Universitat de Girona, Campus Montilivi, Girona, 17003, Spain

166 Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, Tartu, 50411, Estonia

167 Department of Physics and Astronomy, Haverford College, Haverford, PA, 19041, USA

168 Department of Physics, Duke University, Durham, NC, 27708, USA

169 Depto. de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Plaza de las Ciencias 1, Madrid, 28040, Spain

170 Main Astronomical Observatory of National Academy of Sciences of Ukraine, 27 Akademik Zabolotnyi St., Kyiv, 03143, Ukraine

171 Departamento de Matemáticas, Universidad de León, Campus de Vegazana, s/n, León, 24071, Spain

172 Departamento de Física Fundamental and Instituto Universitario de Física Fundamental y Matemáticas (IUFFyM), Universidad de Salamanca, Plaza de la Merced, s/n, Salamanca, E-37008, Spain

173 Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, Potsdam, 14482, Germany

174 Canada-France-Hawaii Telescope, 65-1238 Mamalahoa Hwy, Waimea, HI, 96743, USA

175 School of Physical and Chemical Sciences, University of Canterbury, Private Bag 4800, Christchurch, 8140, New Zealand

176 Centro de Investigaciones Energéticas Medioambientales y Tecnológicas CIEMAT, Av Complutense 40, Madrid, 28040, Spain

177 Physik-Institut, University of Zürich, Winterthurerstrasse 190, Zürich, CH-8057, Switzerland

178 Instituto Tecnológico de Piedras Negras, Mexico

179 LUX, CNRS, Observatoire de Paris, France

180 Instituto de Física Corpuscular (IFIC), University of Valencia-CSIC, Parc Científic UV, c/ Catedrático José Beltrán 2, Paterna, E-46980, Spain

181 Bulgarian Academy of Sciences, Institute for Nuclear Research and Nuclear Energy, 72 Tzarigradsko Chaussee, Sofia, 1784, Bulgaria

182 Department of Physics, Aristotle University of Thessaloniki, A.U.Th. Campus, Thessaloniki, 54635, Greece

183 Departamento de Física Teórica and IPARCOS, Universidad Complutense de Madrid, Madrid, E-28040, Spain

184 INAF - Osservatorio Astronomico di Trieste, via Tiepolo 11, Trieste, 34131, Italy

185 IFPU - Institute for Fundamental Physics of the Universe, via Beirut 2, Trieste, 34151, Italy

186 Center for Astrophysics Harvard & Smithsonian, 60 Garden Street, Cambridge, MA, 02138, USA

187 Black Hole Initiative, Harvard University, 20 Garden Street, Cambridge, MA, 02138, USA

188 STAR Institute, University of Liège, Quartier Agora, Allée du six Août 19c, Liège, 4000, Belgium

189 Department of Theoretical Physics and Condensed Matter Physics (020), Vinča Institute of Nuclear Sciences - National Institute of the Republic of Serbia, University of Belgrade, P.O. Box 522, Belgrade, 11001, Serbia

190 School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran

191 Innovation and Management Research Center, Ayatollah Amoli Branch, Islamic Azad University, Amol, Mazandaran, Iran

192 Physics Department, Michigan Technological University, 1400 Townsend Dr, Houghton, MI, 49931, USA

193 Instituto Nacional de Astrofísica, Optica y Electronica, L.E.Erro N.1, Tonantzintla, Puebla, Mexico

194 Dipartimento di Fisica e Astronomia "Augusto Righti" - Università di Bologna, via Piero Gobetti 93/2, Bologna, I-40129, Italy

195 Department of Physics, College of Science, Kuwait University, Sabah Al Salem University City, P.O. Box 2544, Safat 1320, Kuwait

196 Physics Department, Sapienza University of Rome, P.le A. Moro 5, Roma, 00185, Italy

197 Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Roma, P.le A. Moro 5, Roma, I-00185, Italy

198 Physics Department, Tor Vergata University of Rome, Via della Ricerca Scientifica 1, Roma, 00133, Italy

199 Kayli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo, Chiba, 277-8583, Japan

200 Faculty of Natural Sciences and Mathematics, Department of Mathematics, University of Tuzla, Ul. Urfeta Vejzagića br. 4, Tuzla, 75000, Bosnia and Herzegovina

201 Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, Nijenborgh 3, Groningen, 9747 AG, The Netherlands

202 Department of Physics and Astronomy, Aarhus University, Aarhus C, DK-8000, Denmark

203 Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM, 87131, USA

204 School of Mathematical and Physical Sciences, University of Sheffield, Hounsfield Road, Sheffield, S3 7RH, UK

205 INAF Osservatorio Astronomico d'Abruzzo, via Mentore Maggini snc, Teramo, I-64100, Italy

206 INAF - Osservatorio Astronomico di Roma (OAR), via Frascati 33, Monte Porzio Catone, RM, 00078, Italy

207 Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

208 Department of Physics, Indian Institute of Technology, Guwahati, 781039, India

209 Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI, 96822, USA

210 Istituto Nazionale di Fisica Nucleare, Sez. di Napoli, Via Cinthia 21, Napoli, 80126, Italy

211 Departamento de Física Teórica and IFIC, Centro Mixto Universitat de València - CSIC. Universitat de València, Valencia, Burjassot-46100, Spain

212 Department of Informatics and Telecommunications, University of Peloponnese, Karaiskaki 70, Tripoli, 22100, Greece

213 Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1, Torino, I-10125, Italy

214 INFN - Sezione di Torino, Via P. Giuria 1, Torino, I-10125, Italy

215 INAF - Istituto Nazionale di Astrofisica, Osservatorio Astrofisico di Torino, strada Osservatorio 20, Pino torinese, 10025, Italy

216 Departamento de Física, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, Lisbon, P-1749-016, Portugal

217 Institut de Ciències del Cosmos, Universitat de Barcelona, Martí i Franquès, 1, Barcelona, E-08028, Spain

218 ICREA, Pg. Lluís Companys 23, Barcelona, E-08010, Spain

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

220 Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC, 27599-3255, USA

221 Department of Physics, University of Malta, Msida, Malta

222 Department of Physics and Astronomy, University of California, Davis, CA, USA

223 Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos, 62210, México

224 Departamento de Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280, Antofagasta, Chile

225 Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban, 4000, South Africa
226 Physics Department, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece

227 Faculty of Science and Technology, University of Stavanger, Stavanger, 4036, Norway

228 European Southern Observatory, Karl-Schwarzschild str. 2., Garching, 85748, Germany

229 Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Via Giuseppe Saragat 1, Ferrara, I-44122, Italy

230 Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe Saragat 1, Ferrara, I-44122, Italy

231 INFN - Laboratori Nazionali di Frascati (LNF), Via E. Fermi 54, Frascati, Roma, 00044, Italy

232 National & Kapodistrian University of Athens, Department of Physics, Zografou Campus, Athens, GR 157 73, Greece

233 ENEA, Fusion and Nuclear Safety Department, C.R. Frascati, Via E. Fermi 45, Frascati, 00044, Italy

234 Physics Department, "Sapienza" University of Rome, P.le Aldo Moro 5, Rome, 00185, Italy

235 Departamento de Física, Facultad de Ciencias, Universidad de Tarapacá, Casilla 7-D, Arica, Chile

236 INAF - Osservatorio Astronomico di Roma, Via Frascati 33, Monte Porzio Catone, Roma, I-00040, Italy

237 Dipartimento di Ingegneria, Università LUM, S.S. 100 km 18, Casamassima, Bari, 70010, Italy

238 Department of Chemistry, Physics and Environmental and Soil Sciences, Escola Politécnica Superior, Universidad de Lleida, Av. Jaume II, 69, Lleida, 25001, Spain

239 Universidade Federal do Ceará (UFC), Departamento de Física, Campus do Pici, C.P. 6030, Fortaleza, CE, 60455-760, Brazil

240 Department of Physics, University of Nis, Visegradská 33, Nis, Serbia, SEENET-MTP Centre, Nis, Serbia

241 ESTEC - European Space Agency, Keplerlaan 1, Noordwijk, 2201 AZ, The Netherlands

242 Tsung-Dao Lee Institute (TDLI), No. 1 Lisuo Road, Shanghai, 201210, China

243 School of Physics and Astronomy, Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai, 201240, China

244 Institute for Astronomy, the School of Physics, Zhejiang University, Hangzhou, 310027, People's Republic of China

245 Department of Physics, Southern University of Science and Technology, Shenzhen, 518055, China

246 Department of Physics & Astronomy, University of Kansas, 1251 Wescoe Hall Drive, Lawrence, KS, 66045, USA

247 Institute of Physics, University of Szczecin, Wielkopolska 15, Szczecin, 70-451, Poland

248 Physics Department, Ariel University, Ariel,

249 High Energy Physics Research Unit, Faculty of Science, Chulalongkorn University, Bangkok, 10330, Thailand

250 CEICO, Institute of Physics, Czech Academy of Sciences, Na Slovance 2, Praha, 182 21, Czech Republic

251 Main Astronomical Observatory of the National Academy of Sciences of Ukraine, Akademik Zabolotnyi 27, Kyiv, 03143, Ukraine

252 Department of Astronomy of the University of Geneva, 51 chemin de Pegasi, Versoix, 1290, Switzerland

253 Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, Bilbao, 48080, Spain

254 Shamakhy Astrophysical Observatory, Shamakhy, 5626, Azerbaijan

255 Physics Department, Swansea University, UK

256 Departamento de Física Teórica, Centro de Astropartículas y Física de Altas Energías, Universidad de Zaragoza, Zaragoza, 50009, Spain

257 Institute of Physics, University of Silesia, Katowice, Poland

258 Univ. Lille, CNRS, Centrale Lille, UMR 9189 CRISTAL, Lille, F-59000, France

259 Université Paris-Saclay, CNRS, Institut d'Astrophysique Spatiale, Orsay, 91405, France

260 Helsinki Institute of Physics, P.O. Box 64, University of Helsinki, Helsinki, FI-00014, Finland

261 Academia Sinica Institute of Astronomy and Astrophysics, 11F of AS/NTU Astronomy-Mathematics Building, Roosevelt Rd, Taipei, 106216, Taiwan, ROC

262 Bahamas Advanced Study Institute and Conferences, 4A Ocean Heights, Hill View Circle, Stella Maris, Long Island, The Bahamas

263 Department of Physics & Astronomy, University of Hawaiiʻi at Manoa, Watanabe Hall, 2505 Correa Road, Honolulu, HI, 96822, USA

264 Leung Center for Cosmology & Particle Astrophysics, National Taiwan University, Taipei, 10617, Taiwan

265 National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA, 22903, USA

266 NSF NOIRLab, 950 N. Cherry Avenue, Tucson, AZ, 85719, USA

267 Departamento de Física Teórica and IPARCOS, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Madrid, E-28040, Spain

268 Utah Valley University, Orem, Utah, USA

269 Instituto de Física Teórica UAM/CSIC, Universidad Autonoma de Madrid, Cantoblanco, Madrid, 28049, Spain

270 Nordita, KTH Institute of Technology and Stockholm University, Stockholm, SE-106 91,

271 Department of Physics, National University of Ireland, Maynooth, Ireland

272 Department of Physics, University of Arizona, Tucson, AZ, 85721, USA

273 Department of Physics, University of Maryland, College Park, MD, 20742, USA

274 Leiden Observatory, Leiden University, P.O. Box 9513, Leiden, 2300 RA, The Netherlands

275 Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, UK

276 Department of Theoretical Physics, University of Szeged, Tisza Lajos krt. 84-86, Szeged, H-6720, Hungary

277 Department of Theoretical Physics, HUN-REN Wigner Research Centre for Physics, Konkoly-Thege Miklós út 29-33, Budapest, H-1121, Hungary

278 Institute of Physics, University of Tartu, Estonia

279 Department of Physics and Astronomy, UNC-Chapel Hill, USA

280 Instituto de Física, Universidade Federal Fluminense, Niterói, RJ, 24210-346, Brazil

281 SISSA, Scuola Internazionale Superiore di Studi Avanzati, via Bonomea, 265, Trieste, 34136, Italy

282 Department of Physics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

283 INRNE, Bulgarian Academy of Sciences, Sofia, 1784, Bulgaria

284 Department of Physics, University of Tirana, Boulevard "Zogu I", Tirana, Albania

285 Department of Physics, Shanghai University, Shanghai, 200444, China

286 Asia Pacific Center for Theoretical Physics, Pohang, 37673, Korea

287 CFISUC, Department of Physics, University of Coimbra, Coimbra, P-3004 - 516, Portugal

288 INAF, Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, I-80131 Napoli, Italy & DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, Copenhagen, 2200, Denmark

289 NSF NOIRLab, 950 N Cherry Ave, Tucson, AZ, 85719, USA

290 Astronomy Unit, Queen Mary University of London, Mile End Road, London, E1 4NS, UK

291 Universidad ECCI, Cra. 19 No. 49-20, Bogotá, 111311, Colombia

292 Center for Theoretical Physics, Polish Academy of Sciences, al. Lotników 32/46, Warsaw, 02-668, Poland

293 PDAT Laboratory, Department of Physics, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran

294 Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, 54124, Greece

295 CERIDES- Excellence in Innovation and Technology, European University of Cyprus, Nicosia, 1516, Cyprus

296 Laboratorio de investigación de Cómputo de Física, Facultad de Ciencias Naturales y Exactas, Universidad de Playa Ancha, Subida Leopoldo Carvallo 270, Valparaíso, Chile

297 Laboratorio de Didáctica de la Física, Departamento de Matemática, Física y Computación, Facultad de Ciencias Naturales y Exactas, Universidad de Playa Ancha, Subida Leopoldo Carvallo 270, Valparaíso, Chile

298 University of Wrocław, Institute of Theoretical Physics, pl. Maxa Borna 9, Wrocław, 50-206, Poland

299 University of Torino and INFN, via P. Giuria 1, Torino, 10125, Italy

300 Tartu Observatory, University of Tartu, Observatoorium 1, Tõravere, 61602, Estonia
301 Institute of Physics, University of Tartu, W. Ostwaldi 1, Tartu, 50411, Estonia

302 Physics Department, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece

303 IKERBASQUE, Basque Foundation for Science, Bilbao, 48011, Spain

304 Department of Physics & EHU Quantum Center, University of the Basque Country UPV/EHU, P.O. Box 644, Bilbao, 48080, Spain

305 Laboratoire d'étude de l'Univers et des phénomènes extrêmes, Observatoire de Paris, UMR 8262 CNRS, Sorbonne Université,

306 High Energy Physics Division, Argonne National Laboratory, Lemont, IL, 60439, USA

307 Universe-Origins, University of Southern Denmark, Campusvej 55, Odense M, 5230, Denmark

308 Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study (UTIAS), The University of Tokyo, Chiba, 277-8583, Japan

309 Department of Physics, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran

310 Dipartimento di Fisica e Astronomia "A. Righi" - Alma Mater Studiorum Università di Bologna, via Piero Gobetti 93/2, 40129 Bologna, Italy

311 INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Piero Gobetti 93/3, 40129, Bologna, Italy

312 Technical University Munich, School of Natural Sciences, Physik Department T31, James-Franck Str. 1, Garching, 85748, Germany

313 Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Física, Buenos Aires, Argentina

314 CONICET - Universidad de Buenos Aires, Instituto de Física de Buenos Aires (IFIBA), Buenos Aires, Argentina

315 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, Monteporzio Catone, 00078, Italy

316 INFN-Sezione di Roma, Piazzale Aldo Moro, 2 - c/o Dipartimento di Fisica, Edificio G. Marconi, Roma, 00185, Italy

317 Dipartimento di Fisica e Astronomia Augusto Righi - Università di Bologna, via Piero Gobetti 93/2, Bologna, I-40129, Italy

318 Department of Physics, Karadeniz Technical University, Trabzon, TR61080, Türkiye

319 Depto. de Física y Matemáticas, Universidad Iberoamericana Ciudad de México, Prolongación Paseo de la Reforma 880, México D. F., 01219, Mexico

320 Faculty of Sciences and Mathematics, University on Nis, Serbia

321 Konkoly Observatory, HUN-REN Research Centre of Astronomy and Earth Sciences (CSFK), MTA Centre of Excellence, Konkoly Thege Miklós út 15-17., Budapest, H-1121, Hungary

322 Department of Physics, The University of Texas at Dallas, Richardson, TX, 75080, USA

323 Astronomical Institute, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, Praha, CZ-180 00, Czech Republic

324 Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstr. 1, München, D-81679, Germany

325 INFN-Sezione di Bologna, Viale Berti Pichat 6/2, Bologna, 40127, Italy

326 Laboratoire des 2 Infinis - Toulouse (L2IT-IN2P3), Université de Toulouse, CNRS, Toulouse Cedex 9, F-31062, France

327 University of Torino and INFN/Torino, via P. Giuria 1, Torino, 10125, Italy

328 Department of Electrical and Electronics Engineering Doğuş University, Umraniye, Istanbul, 34775, Türkiye

329 Physics Department, King's College London, Strand, London, WC2R 2LS, UK

330 Department of Physics, Faculty of Science, University of Thessaly, Thessaly, GR 35100, Greece

331 Duke University, Durham, NC, 27708, USA

332 Cosmology, Gravity and Astroparticle Physics Group, Center for Theoretical Physics of the Universe, Institute for Basic Science, Daejeon, 34126, Korea

333 LTE, Observatoire de Paris, Université PSL, CNRS, LNE, Sorbonne Université, 61 avenue de l'Observatoire, Paris, 75 014, France

334 Department of Physics, Sharif University of Technology, Tehran, 11155-9161, Iran

335 Department of Physics, Shahid Beheshti University, Tehran, 1983969411, Iran

336 Center for Research and Training in Innovative Techniques of Applied Mathematics in Engineering, University Politehnic a of Bucharest, Bucharest, 060042, Romania

337 International Centre for Space and Cosmology, Ahmedabad University, Ahmedabad, 380009, India

338 Department of Physics and Astronomy, Vanderbilt University, Nashville, TN, 37235, USA

339 Main Astronomical Observatory of the National Academy of Sciences of Ukraine, 27 Akademik Zabolotny St., Kyiv, 03143, Ukraine

340 Astronomical Observatory, Taras Shevchenko National University of Kyiv, 3 Observatorna St., Kyiv, 04053, Ukraine

341 Department of Physics, University of Aberdeen, Aberdeen, AB24 3UE, UK

342 Department of Physics, University of the Basque Country, Spain

343 INFN Sezione di Ferrara, Via Saragat 1, Ferrara, 44122, Italy

344 ICSC, Centro Nazionale "High Performance Computing, Big Data and Quantum Computing", Italy

345 Research School of Astronomy and Astrophysics, The Australian National University, Mount Stromlo Observatory, Canberra, ACT, 2611, Australia

346 Department of Astronomy, Vesilinnantie 5, University of Turku, Turku, 20014, Finland

347 Centre for Theoretical Cosmology, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

348 Department of Physics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, 9 Iroon Polytechniou Str., Zografou Campus, Athens, GR 157 80, Greece

349 Department of Physics, Northeastern University, Boston, MA, 02115, USA

350 Astronomical Observatory, Volgina 7, P.O. Box 74, Belgrade, 11060, Serbia

351 Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata, 700 108, India

352 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 155, Copenhagen, 2200, Denmark

353 Institute of Physics, Laboratory of Astrophysics, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, Versoix, CH-1290, Switzerland

354 Department of Physics, Ankara University, Faculty of Sciences, Ankara, 06100, Turkiye

355 INAF - Osservatorio Astronomico d'Abruzzo, Via Maggini, Teramo, 64100, Italy

356 Physics Department, Mapúa University, 658 Muralla St., Intramuros, Manila, 1002, Philippines

357 Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK

358 School of Physics and Astronomy, University of Southampton, Southampton, UK

359 Instituto Nacional de Astrofísica, Óptica y Electrónica, Tonantzintla, Puebla, México

360 Facultad de Astronomía y Geofísica, Universidad de La Plata, La Plata, Argentina

361 INFN - Sezione di Roma 1, P.le Aldo Moro 2, Roma, 00185, Italy

362 Instituto de Astronomía, Universidad Nacional Autónoma de México, Av. Universidad 3000, Col. Universidad Nacional Autónoma de México, Ciudad de México, C.P. 04510,

363 Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte, Campus Universitário Lagoa Nova, Natal, RN, 59078-970, Brazil

364 Physics Department, University of the Basque Country UPV/EHU, 644 PO Box, Bilbao, 48080, Spain

365 Department of Physics, University of Alberta, Edmonton, Alberta, T6G 2E1, Canada

366 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada

367 Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, T6G 2E1, Canada

368 Islamic Azad University, Science and Research Branch, Tehran, Iran

369 Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Ferrara, 44122, Italy

370 Dipartimento di Fisica, Università di Trento, Trento, 38123, Italy

371 Department of Physics, Columbia University, New York, NY, USA

372 SISSA, Via Bonomea 265, Trieste, 34136, Italy

373 Institute for Fundamental Physics of the Universe (IFPU), Via Beirut 2, Trieste, 34014, Italy
³⁷⁴ Faculty of Symbiotic Systems Science, Fukushima University, Fukushima, 960-1296, Japan

³⁷⁵ Theoretical Physics Group and Quantum Alberta, Department of Physics and Astronomy, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, T1K 7Z2, Canada

³⁷⁶ Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study (UTIAS), The University of Tokyo, Kashiwa, Chiba, 277-8583, Japan

³⁷⁷ Cosmology, Gravity, and Astroparticle Physics Group, Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), Daejeon, 34126, Korea

³⁷⁸ Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstrasse 25, Innsbruck, 6020, Austria

³⁷⁹ Institute of Space Sciences (ICE, CSIC), C. Can Magrans s/n, Barcelona, 08193, Spain

³⁸⁰ ICREA, Passeig Lluis Companys, 23, Barcelona, 08010, Spain

³⁸¹ Universität Bonn, Regina-Pacis-Weg 3, Bonn, 53113, Germany

³⁸² Institute of Cosmology & Gravitation, Dennis Sciama Building, University of Portsmouth, Portsmouth, PO1 3FX, UK

³⁸³ Technical University of Munich, TUM School of Natural Sciences, Physics Department, James-Franck-Straße 1, Garching, 85748, Germany

³⁸⁴ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild Straße 1, Garching, 85748, Germany

³⁸⁵ Institute of Astronomy and Astrophysics, Academia Sinica, No. 1, Section 4, Roosevelt Road, Taipei, 106319, Taiwan

³⁸⁶ Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing, Bochum, 44780, Germany

³⁸⁷ Leiden Observatory, Leiden University, Einsteinweg 55, Leiden, 2333 CC, The Netherlands

³⁸⁸ University of California, Riverside, CA, 92507, USA

³⁸⁹ Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY, 11794, USA

³⁹⁰ INAF - Osservatorio di astrofisica e scienza dello spazio, via Gobetti 93/3, Bologna, 40129, Italy

³⁹¹ Dipartimento di Fisica, Università di Trento, Via Sommarive 14, Povo, Trento, 38123, Italy

³⁹² Trento Institute for Fundamental Physics and Applications (TIFPA), Via Sommarive 14, Povo, Trento, 38123, Italy

³⁹³ Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse 1, Garching, 85748, Germany

³⁹⁴ Universite Paris-Saclay, CNRS/IN2P3, IJCLLab, Orsay, 91405, France

³⁹⁵ Academy of Athens, Research Center for Astronomy and Applied Mathematics, Soranou Efesiou 4, Athens, 11527, Greece

³⁹⁶ School of Sciences, European University Cyprus, Diogenes Street, Engomi, Nicosia, 1516, Cyprus

³⁹⁷ Università di Torino, Physics department, via P. Giuria 1, Turin, 10125, Italy

³⁹⁸ Department of Physics, University of Thessaly, Lamia, 35100, Greece

³⁹⁹ Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, Warszawa, PL-00478, Poland

⁴⁰⁰ Data Science Institute, Plaksha University, Mohali, Punjab, 140306, India

⁴⁰¹ CONICET-Universidad de Buenos Aires-Instituto de Física de Buenos Aires, Av. Intendente Cantilo S/N, CABA, 1428, Argentina

⁴⁰² International Gemini Observatory, NSF NOIRLab, Casilla 603, La Serena, Chile

⁴⁰³ Laboratorio Nacional de Astrofísica - MCTI, Itajuba, MG, 37504-364, Brazil

⁴⁰⁴ Center for Astrophysics and Cosmology, University of Nova Gorica, Vipavska 11c, Ajdovščina, 5270, Slovenia

⁴⁰⁵ Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Via G. Saragat 1, Ferrara, I-44122, Italy

⁴⁰⁶ Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Ferrara, Via G. Saragat 1, Ferrara, I-44122, Italy

⁴⁰⁷ Institute of Space Sciences (ICE, CSIC), C. Can Magrans s/n, Barcelona, 08193, Spain

⁴⁰⁸ Galileo Galilei Institute for Theoretical Physics, Largo Enrico Fermi 2, Firenze, I-50125, Italy

⁴⁰⁹ Department of Physics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

⁴¹⁰ National Institute of Chemical Physics and Biophysics, Rävala pst. 10, Tallinn, 10143, Estonia

⁴¹¹ Department of Physics, University of Oslo, Box 1048, Oslo, N-0316, Norway

⁴¹² Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, Bellaterra (Barcelona), 08193, Spain

⁴¹³ Department of Physics, University of Ottawa, 75 Laurier Ave E, Ottawa, Canada

⁴¹⁴ Astrophysics Research Center of the Open University, The Open University of Israel, Raanana, Israel

⁴¹⁵ Departamento de Física, Universidade Federal do Espírito Santo, Vitória, ES, 29075-910, Brazil

⁴¹⁶ Department of Physics and Astronomy, Sejong Universe, Seoul, 05006, South Korea

⁴¹⁷ INAF-OAS, Bologna, via P. Gobetti 101, I-40129, Italy

⁴¹⁸ Scottish Universities Physics Alliance, University of Saint Andrews, North Haugh, Saint Andrews, Fife, KY16 9SS, UK

⁴¹⁹ Instituto de Física Corpuscular (IFIC), CSIC - Universitat de València, C/ Catedrático José Beltrán 2, Paterna (Valencia), 46980, Spain

⁴²⁰ Department of Mathematics, University of Thessaly, GR-35100, Greece

⁴²¹ Department of Mathematics, Faculty of Natural and Mathematical Sciences, University of Tuzla, Urfeta Vezjagica 4, Tuzla, 75000, Bosnia and Herzegovina

⁴²² School of Theoretical Physics, Dublin Institute for Advanced Studies (DIAS), 10 Burlington Road, Dublin, D04 C932, Ireland

⁴²³ Department of Physics and Helsinki Institute of Physics, Gustaf Hallström katu 2, University of Helsinki, Helsinki, FIN-00014, Finland

⁴²⁴ Instituto de Física y Astronomía, Universidad de Valparaíso, Avda. Gran Bretaña 1111, Valparaíso, Chile

⁴²⁵ Leiden Observatory, Leiden University, PO Box 9513, Leiden, 2300 RA, The Netherlands

⁴²⁶ Max-Planck-Institut für extraterrestrische Physik, Gießenbachstraße 1, Garching, 85748, Germany

⁴²⁷ INAF, Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3, Bologna, 40129, Italy

⁴²⁸ Center for Physical Sciences and Technology, Saulėtekio av. 3, Vilnius, 10257, Lithuania

⁴²⁹ Department of Physics, Liaoning Normal University, Dalian, 116029, P. R. China

⁴³⁰ Department of Astronomy, School of Physical Sciences, University of Science and Technology of China, Hefei, 230026, China

⁴³¹ Department of Physics, Institute of Science Tokyo, Tokyo, 152-8551, Japan

⁴³² CAS Key Laboratory for Research in Galaxies and Cosmology, School of Astronomy and Space Science, University of Science and Technology of China, Hefei, 230026, China

⁴³³ Department of Mathematics, University of Okara, Okara, 56300, Pakistan

⁴³⁴ Center for Astronomy and Astrophysics, Center for Field Theory and Particle Physics and Department of Physics, Fudan University, Shanghai, 200438, China

⁴³⁵ Institute of Physics and Astronomy, ELTE Eötvös Loránd University, Budapest, 1117, Hungary

⁴³⁶ Facultad de Física, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago, Chile

⁴³⁷ Institut für Theoretische Physik und Atominstitut, Technische Universität Wien, Wiedner Hauptstrasse 8–10, Vienna, A-1040, Austria

⁴³⁸ Indian Institute of Astrophysics, Bengaluru, Karnataka, 560034, India

⁴³⁹ University of Wroclaw, Poland

⁴⁴⁰ Complutense University of Madrid,

⁴⁴¹ Max Planck Institute for Radio Astronomy, Auf dem Huegel 69, Bonn, D-53121, Germany

⁴⁴² Institute of Physics, Silesian University in Opava, Opava, CZ-74601, Czech Republic

⁴⁴³ Universiti Malaya, Kuala Lumpur, Wilayah Persekutuan Kuala Lumpur, 50603,

⁴⁴⁴ Department of Physics, Tezpur University, Napaam, Tezpur, Assam, 784028, India

⁴⁴⁵ Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK

⁴⁴⁶ Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK

⁴⁴⁷ Higgs Centre for Theoretical Physics, School of Physics and Astronomy, Edinburgh, EH9 3FD, UK

⁴⁴⁸ Department of Mathematics, BITS-Pilani Hyderabad Campus, India
<footnote n="449">449 Institute for Advanced Studies, New Uzbekistan University, Movarouonnastr. 1, Tashkent, 100000, Uzbekistan</footnote>

<footnote n="450">450 Institute of Theoretical Physics, National University of Uzbekistan, Tashkent, 100174, Uzbekistan</footnote>

<footnote n="451">451 Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile,</footnote>

<footnote n="452">452 The Oskar Klein Centre, Department of Physics, Stockholm University, Stockholm, SE - 106 91, Sweden</footnote>

<footnote n="453">453 Department of Physics, Department of Astronomy & Astrophysics, KICP, and EFI, University of Chicago, Chicago, IL, 60637, USA</footnote>

<footnote n="454">454 Institute of Theoretical Astrophysics, University of Oslo, Oslo, 0315, Norway</footnote>

<footnote n="455">455 Department of Physics and Astronomy, PAB, 430 Portola Plaza, Box 951547, Los Angeles, CA, 90095-1547, USA</footnote>

<footnote n="456">456 Departments of Astronomy and Physics, Boston University, Boston, MA, 02215, USA</footnote>

<footnote n="457">457 Ben Gurion University of the Negev, Beer Sheva, Israel</footnote>

<footnote n="458">458 Departamento de Física Teórica, Módulo 15, Facultad de Ciencias, Universidad Autónoma de Madrid, Madrid, 28049, Spain</footnote>

<footnote n="459">459 Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, 91501-970, Brazil</footnote>

<footnote n="460">460 Department of Physics, Faculty of Arts and Sciences, Istanbul Technical University, Istanbul, Turkey</footnote>

<footnote n="461">461 Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, UK</footnote>

<footnote n="462">462 University of Zurich, Winterthurerstrasse 190, Zurich, 8057, Switzerland</footnote>

<footnote n="463">463 Instituto Superior Tecnico - IST, Universidade de Lisboa, Lisboa, 1049-001, Portugal</footnote>

<footnote n="464">464 Laboratorio de Instrumentaycao e Fisica Experimental de Partulas, 1649-003 Lisboa and 3004-516 Coimbra, Portugal</footnote>

<footnote n="465">465 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, Bonn, 53121, Germany</footnote>

<footnote n="466">466 Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM, 87106, USA</footnote>

<footnote n="467">467 Heidelberg Institute for Theoretical Studies (HITS), Schloss-Wolfsbrunnenweg 35, Heidelberg, 69118, Germany</footnote>

<footnote n="468">468 Department of Physics, University of Helsinki, Gustaf Hällströmin katu 2, Helsinki, FI-00014, Finland</footnote>

<footnote n="469">469 National Institute of Physics, University of the Philippines, Quezon City, Philippines</footnote>

<footnote n="470">470 Sydney Institute for Astronomy, School of Physics, A28, The University of Sydney, NSW, 2006, Australia</footnote>

<footnote n="471">471 L2IT, Laboratoire des 2 Infinis - Toulouse, CNRS/IN2P3, UPS, Toulouse Cedex 9, F-31062, France</footnote>

<footnote n="472">472 MTA-ELTE Astrophysics Research Group, Budapest, 1117, Hungary</footnote>

<footnote n="473">473 Centro de Astrofísica e Gravitação - CENTRA, Departamento de Física, Instituto Superior Técnico - IST, Universidade de Lisboa - UL, Av. Rovisco Pais 1, Lisboa, 1049-001, Portugal</footnote>

<footnote n="474">474 Dipartimento di Fisica e Astronomia, Università di Firenze, via G. Sansone 1, Sesto Fiorentino, Firenze, 50019, Italy</footnote>

<footnote n="475">475 INAF Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, Firenze, I-50125, Italy</footnote>

<footnote n="476">476 GRAPPA Institute, Institute for Theoretical Physics Amsterdam, Science Park 904, Amsterdam, 1098 XH, The Netherlands</footnote>

<footnote n="477">477 Université Claude Bernard Lyon 1, IUF, IP21 Lyon, 4 rue Enrico Fermi, Villeurbanne, 69622, France</footnote>

<footnote n="478">478 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran</footnote>

<footnote n="479">479 Instituto de Física, Universidade Federal Fluminense, Avenida General Milton Tavares de Souza s/n, Gragoatá, Niterói, Rio de Janeiro, 24210-346, Brazil</footnote>

<footnote n="480">480 Centro de Astrofísica e Gravitação - CENTRA, Departamento de Física, Instituto Superior Técnico - IST, Universidade de Lisboa - UL, Av. Rovisco Pais 1, 1049-001, Lisboa, Portugal</footnote>

<footnote n="481">481 Department of Physics of Complex Systems, ELTE Eötvös Loránd University, Budapest, 1117, Hungary</footnote>

<footnote n="482">482 Department of Physics, Boston University, Boston, Massachusetts, 02215, USA</footnote>

<footnote n="483">483 Institute of Physics, Ss Cyril and Methodius University in Skopje, North Macedonia</footnote>

<footnote n="484">484 Departamento de Física, Universidad de Córdoba, Córdoba, E-14071, Spain</footnote>

<footnote n="485">485 Department of Mathematics, Faculty of Science, University of Malta,</footnote>

<footnote n="486">486 Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, Aachen, 52056, Germany</footnote>

<footnote n="487">487 School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China</footnote>

<footnote n="488">488 Physics Department, State University of Tetovo, Ilinden Street nn, Tetovo, 1200, North Macedonia</footnote>

<footnote n="489">489 Institute for Theoretical Physics, Universität Heidelberg, Philosophenweg 16, Heidelberg, 69120, Germany</footnote>

<footnote n="490">490 Kapteyn Astronomical Institute, University of Groningen, P.O.Box 800, Groningen, 9700AV, The Netherlands</footnote>

<footnote n="491">491 Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad, 500078, India</footnote>

<footnote n="492">492 Département de Physique Théorique, Université de Genève, 24 quai Ansermet, Genève, 1211 4, Switzerland</footnote>

<footnote n="493">493 IATE-OAC-UNC and CONICET, Laprida 854, Barrio Observatorio, Cordoba, 5000, Argentina</footnote>

<footnote n="494">494 Center for Cosmology and Astrophysics, Alikhanian National Laboratory and Yerevan State University, Yerevan, Armenia</footnote>

<footnote n="495">495 Institute of Applied Sciences, The Malta College of Arts, Science and Technology, Paola, Malta</footnote>

<footnote n="496">496 Department of Theoretical Physics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, 84248, Slovak Republic</footnote>

<footnote n="497">497 Centre for Theoretical Physics and Natural Philosophy, Mahidol University, Nakhonsawan Campus, Phayuha Khiri, Nakhonsawan, 60130, Thailand</footnote>

<footnote n="498">498 Javakhishvili Tbilisi State University, 3 Chavchavadze Avenue, Tbilisi, 0179, Georgia</footnote>

<footnote n="499">499 Institut für Theoretische Physik, Universität Münster, Münster, 48149, Germany</footnote>

<footnote n="500">500 Dipartimento di Fisica e Astronomia, Università di Bologna, via Irnerio 46, Bologna, 40126, Italy</footnote>

<footnote n="501">501 INFN, Sezione di Bologna, viale Berti Pichat 6/2, Bologna, 40127, Italy</footnote>

<footnote n="502">502 Instituto de Física & Observatório do Valongo, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil</footnote>

<footnote n="503">503 PPG Cosmo, Universidade Federal do Espírito Santo, Vitória, ES, 29075-910, Brazil</footnote>

<footnote n="504">504 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Muhlenberg 1, Potsdam-Golm, D-14476, Germany</footnote>

<footnote n="505">505 Astronomical Observatory, Volgina 7, Belgrade, 11060, Serbia</footnote>

<footnote n="506">506 Vinca Institute of Nuclear Sciences - National Institute of the Republic of Serbia, University of Belgrade,</footnote>

<footnote n="507">507 Center for Physical Sciences and Technology, Sauletekio av. 3, Vilnius, 10257, Lithuania</footnote>

<footnote n="508">508 Institute of Theoretical Physics & Research Center of Gravitation; Key Laboratory of Quantum Theory and Applications of MoE; and Lanzhou Center for Theoretical Physics & Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, 730000, China</footnote>

<footnote n="509">509 Instituto de Astrofísica e Ciências do Espaço, Universidade de Lisboa, Faculdade de Ciências, Ed. C8, Campo Grande, Lisboa, 1769-016, Portugal</footnote>

<footnote n="510">510 Université de Paris-Cité, APC-Astroparticule et Cosmologie (UMR-CNRS 7164), Paris, F-75006, France</footnote>

<footnote n="511">511 Department of Physical Foundations of Engineering, Politehnicu University of Timisoara, Timisoara, 300223, Romania</footnote>

<footnote n="512">512 National Research University TIAME, Kori Niyoziy 39, Tashkent, 100000, Uzbekistan</footnote>

<footnote n="513">513 Dipartimento di Scienza e Alta Tecnologia (DiSAT), Università degli Studi dell'Insubria, via Valleggio 11, Como, 22100, Italy</footnote>

<footnote n="514">514 Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Milano, via Celoria 16, Milano, 20126, Italy</footnote>

<footnote n="515">515 Como Lake centre for AstroPhysics (CLAP), DiSAT, Università dell'Insubria, via Valleggio 11, Como, 22100, Italy</footnote>

<footnote n="516">516 Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Rio de Janeiro, RJ, 22290-180, Brazil</footnote>

<footnote n="517">517 HUN-RELENTE Extragalactic Astrophysics Research Group, Budapest, 1117, Hungary</footnote>

<footnote n="518">518 School of Science, Constructor University, Bremen, 28759, Germany</footnote>

<footnote n="519">519 Instituto de Física Fundamental (CSIC), Serrano 123, 28006 Madrid, Spain and Institute of Cosmos Science (ICCUB), Martí i Franques 1, Barcelona, 08028, Spain</footnote>

<footnote n="520">520 Universidade Federal do Ceará (UFC), Departamento de Física, Campus do Pici, C.P. 6030, Fortaleza, CE, 60455-760, Brazil</footnote>

<footnote n="521">521 Département de Physique Théorique, Université de Genève, Quai E. Ansemet 24, Genève, 1211, Switzerland</footnote>

<footnote n="522">522 Department of Mathematics, School of Computer Science and Artificial Intellegence, S R Univrsity, Warangal, Telangana, 506371, India</footnote>

<footnote n="523">523 Dipartimento di Fisica e Astronomia G. Galilei, Università degli Studi di Padova, Padova, Italy</footnote>
## ABSTRACT

The standard model of cosmology has provided a good phenomenological description of a wide range of observations both at astrophysical and cosmological scales for several decades. This concordance model is constructed by a universal cosmological constant and supported by a matter sector described by the standard model of particle physics and a cold dark matter contribution, as well as very early-time inflationary physics, and underpinned by gravitation through general relativity. There have always been open questions about the soundness of the foundations of the standard model. However, recent years have shown that there may also be questions from the observational sector with the emergence of differences between certain cosmological probes. In this White Paper, we identify the key objectives that need to be addressed over the coming decade together with the core science projects that aim to meet these challenges. These discordances primarily rest on the divergence in the measurement of core cosmological parameters with varying levels of statistical confidence. These possible statistical tensions may be partially accounted for by systematics in various measurements or cosmological probes but there is also a growing indication of potential new physics beyond the standard model. After reviewing the principal probes used in the measurement of cosmological parameters, as well as potential systematics, we discuss the most promising array of potential new physics that may be observable in upcoming surveys. We also discuss the growing set of novel data analysis approaches that go beyond traditional methods to test physical models. These new methods will become increasingly important in the coming years as the volume of survey data continues to increase, and as the degeneracy between predictions of different physical models grows. There are several perspectives on the divergences between the values of cosmological parameters, such as the model-independent probes in the late Universe and model-dependent measurements in the early Universe, which we cover at length. The White Paper closes with a number of recommendations for the community to focus on for the upcoming decade of observational cosmology, statistical data analysis, and fundamental physics developments.

<figure><img src="image_4.png" /></figure>

## Executive summary

The CosmoVerse network is born out of the CosmoVerse COST Action (formally CA21136 - Addressing observational tensions in cosmology with systematics and fundamental physics [1,2]), which traces its origins to the SNOWMASS 2021 effort in the cosmic tensions sector [3-7]. This will be one of the key deliverables of the COST Action and one of its lasting legacies. The aim of this network is to establish synergy among researchers working across the disparate themes of

observational cosmology, novel techniques of data analysis, and fundamental physics. This White Paper demonstrates how this effort has been successful, while also laying out a roadmap to sustain those efforts and expand the interdisciplinary nature of the topic across different areas of the community. Another core aspect of this effort is to harmonize approaches between the constituent subcommunities and to create a common language in which to discuss the topic of cosmic tensions.

In the White Paper, the accomplishment of these goals is demonstrated through the interwoven connections between the key communities of the network and the granular topics. This was achieved through a diverse set of actions, including the CosmoVerse Seminar

<footnote n="524">524 Institute for Astronomy, University of Edinburgh, Royal Observatory Edinburgh, Blackford Hill, Edinburgh, EH9 3HJ, UK</footnote>

<footnote n="525">525 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Lisboa, PT1749-016, Portugal</footnote>

<footnote n="526">526 Department of Physics, Syracuse University, Syracuse, NY, 13244, USA</footnote>

<footnote n="527">527 Department of Physics and Astronomy, University of South Carolina, Columbia, SC, 29208, USA</footnote>

<footnote n="528">528 Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China</footnote>

<footnote n="529">529 Asia Pacific Center for Theoretical Physics (APCTP), Pohang, 37673, Korea</footnote>

<footnote n="530">530 Department of Physics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tiruvalam Rd, Katpadi, Tamil Nadu, 632014, India</footnote>

<footnote n="531">531 Department of Astronomy & Astrophysics, Tata Institute of Fundamental Research, 1, Homi Bhabha Road, Colaba, Mumbai, 400005, India</footnote>

<footnote n="532">532 Ruer Bokovi Institute, Zagreb, Croatia</footnote>

<footnote n="533">533 Institute of Cosmology and Gravitation, University of Portsmouth,</footnote>

<footnote n="534">534 ECEO, Universidade Lusófona, Campo Grande 376, Lisboa, 1749-024, Portugal</footnote>

<footnote n="535">535 Department of Physics and Materials Science, University of Luxembourg, Luxembourg City, L-1511, Luxembourg</footnote>

<footnote n="536">536 Astronomical Observatory Belgrade, Volgina 7, Belgrade, 11060, Serbia</footnote>

<footnote n="537">537 Mathematical Sciences and STAG Research Centre, University of Southampton, Southampton, SO17 1BJ, United Kingdom</footnote>

<footnote n="538">538 Institute of physics Belgrade, Belgrade University, Pregrevica 118, Belgrade, Serbia</footnote>

<footnote n="539">539 Department of Physics, the Chinese University of Hong Kong, Sha Tin, NT, Hong Kong,</footnote>

<footnote n="540">540 Department of Physics, Hokkaido University, Sapporo, 060-0810, Japan</footnote>

<footnote n="541">541 School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou, 310024, China</footnote>

<footnote n="542">542 International Center for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China</footnote>

<footnote n="543">543 School of Physics, Korea Institute for Advanced Study (KIAS), 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Korea</footnote>
Series led by Eleonora Di Valentino [8], the CosmoVerse conferences and the discussions and presentations involved in these events [9], the CosmoVerse School held in Corfu in 2024 [10], the CosmoVerse Training Series, which involved a significant effort to bridge [11], as well as the CosmoVerse Journal Club, led by Enrico Specogna and Mahdi Najafi [12].

The CosmoVerse White Paper, edited by Jackson Levi Said and Eleonora Di Valentino, is a collective effort from the community to review the state of the art and identify opportunities to address tensions in cosmology over the coming years. This includes progress in observational cosmology and the exhaustive treatment of potential systematics, the development of new data analysis approaches for upcoming surveys and potential new physics models, as well as the construction of cosmological models that appropriately address the areas where the concordance model exhibits tensions. These topics are organized as follows:

* Section 1 - Introduction and state-of-the-art.

* Section 2 - Observational cosmology. Coordinators: Adam Riess, Amanda Weltman, Antonella Palmese, Caroline D. Huang, Chandra Shekhar Saraf, Cheng-Yu Kuo, Cora Uhlemann, Dan Scolnic, Daniela Grandón, Dante Paz, Dominique Eckert, Florian Beutler, Gabriela Barenboim, Ilaria Musella, Istvan Szapudi, Jack Singal, Khaled Said, Leandros Perivolaropoulos, Lluís Galbany, Louise Breuval, Louise Breuval, Maria Giovanna Dainotti, Maria Vincenzi, Marika Asgari, Martina Gerbino, Margherita Lembo, Matteo Forconi, Michele Cantiello, Michele Moresco, Nils Schöneberg, Ricardo Chávez Murillo, Richard I. Anderson, Rick Watkins, Shahin Sheikh-Jabbari, Siyang Li, Tommaso Treu, Vid Iršič, Will Handley, William Giarè and Yukei Murakami.

* Section 3 - Novel data analysis techniques. Coordinators: Agnieszka Pollo, Adriá Gómez-Valent, Daniela Grandón, Jesus Torrado, Jurgen Mifsud, Lei Zu, Luis Escamilla, and Reginald Christian Bernardo.

* Section 4 - Fundamental physics. Coordinators: Vivian Poulin, Carsten van de Bruck, Elsa Teixeira, Emmanuel N. Saridakis, Eoin O Colgain, Florian Niedermann, Francesco Bajardi, Giulia Gubitosi, In-dranil Banik, Jaime Haro Cases, Jens Chluba, Karsten Jedamzik, Kon-stantinos F. Dialektopoulos, Laura Herold, Leandros Perivolaropoulos, Luca Visinelli, Luis Anchordoqui, Micol Benetti, Özgür Akarsu, Rafael Nunes, Sunny Vagnozzi, Supriya Pan.

* Section 5 - Discussion, White Paper recommendations, and future prospects.

In every section, the coordinators and contributing writers are identified at the beginning of each contribution. The preparation of the CosmoVerse White Paper involved a substantial number of people, with ~65 coordinators, ~350 contributing writers, and ~130 endorsers. The project was reviewed by Alba Domi, Alexandra Amon, Anjitha John William Mini Latha, Anton Chudaykin, Bivudutta Mishra, Emil Mottola, Emmanuel N. Saridakis, Florian Pacaud, Germano Nardini, Marcin Postolak, Mariano Domínguez, Miguel A. García-Aspeitia, Nelson J. Nunes, Oem Trivedi, Oliver Fabio Piattella, Özgür Akarsu, Paolo Salucci, Pilar Ruiz-Lapuente, Rafid H. Dejrah, and Supriya Pan, and the bibliographic information was organized by Luca Visinelli. The notational conventions of the work are defined in Section 6. This is followed by a list of the glossary abbreviations in Section 7. The CosmoVerse Action and White Paper were aided by the administrative contributions of Hiba Wazaz.

# 1. Introduction

Cosmology has entered an era of unprecedented precision in and volume of observational measurements, with large-scale surveys providing high-quality data across multiple redshifts and cosmic epochs. This wealth of observational data allows for a deeper understanding of the Universe's composition, dynamics, and structure formation

processes of the Universe. However, these advancements have also revealed significant tensions between early- and late-time cosmological measurements, challenging the standard cosmological model. These discrepancies not only question the consistency of this framework but also suggest the possibility of unrecognized systematic errors or the need for new physics beyond the standard model of cosmology.

The standard model of cosmology, or ΛCDM describes the Universe using a cosmological constant (Λ) and Cold Dark Matter (CDM), and gravity through Einstein's GR. It provides an excellent fit for a range of cosmological datasets, including the CMB and large-scale structure surveys. Nevertheless, the emergence of tensions in key cosmological parameters has become increasingly difficult to ignore. Among these, the most prominent include discrepancies in the measurements of the Hubble constant ($H_0$), the amplitude of matter fluctuations ($S_8$), and the sound horizon at the epoch of BAO.

These tensions raise profound questions about our understanding of the Universe's expansion history, structure formation, and the fundamental nature of DM and DE. If not due to systematic errors, such tensions may indicate the need for modifications of the standard model or additional components in the cosmic inventory. Addressing these issues requires a careful consideration of both observational and theoretical aspects, as well as a comprehensive approach that combines multiple cosmological probes.

## 1.1. State of the art status of cosmological tensions

The most statistically significant tension in cosmology is the $H_0$ tension, which refers to a significant and persistent discrepancy between measurements of the Hubble constant obtained from early- and late-time cosmological probes, challenging the completeness of the standard ΛCDM model and suggesting the possible need for new physics. Early Universe constraints, primarily from the Planck satellite (Section 2.1.18), which maps CMB anisotropies, provide a precise estimate of $H_0 = 67.4 \pm 0.5 \text{ km s}^{-1}\text{Mpc}^{-1}$. This result relies on the angular scale of the acoustic peaks in the CMB power spectrum, under the assumptions of a flat ΛCDM model with standard radiation content. Consistency with Planck's result has also been demonstrated by ground-based experiments such as ACT and SPT, both of which measure the damping tail and lensing-induced smoothing of the CMB power spectrum with high precision, reinforcing the lower $H_0$ value.

BAO (see Section 2.1.19) provide an additional, independent probe of the expansion history by measuring the characteristic scale imprinted by sound waves in the early Universe, observable in the large-scale distribution of galaxies. This standard ruler, linked to the sound horizon at the epoch of baryon drag, enables precise distance measurements at various redshifts. Surveys such as BOSS, eBOSS, and DESI have reported $H_0$ values consistent with CMB constraints, further supporting the lower early Universe based estimate. Notably, DESI's latest BAO results, based on over six million galaxies across multiple redshift bins, when calibrated by the Planck + ΛCDM constraint on the sound horizon measured $H_0 = 68.5 \pm 0.6 \text{ km s}^{-1}\text{Mpc}^{-1}$.

Late-time measurements of the Hubble constant $H_0$ using the distance ladder approach (Section 2.1.1) suggest a higher value than early Universe constraints, contributing to the persistent Hubble tension. The most precise distance ladder method involves three steps: (1) geometric distance measurements to calibrate the luminosities of Cepheid variables using Gaia parallaxes, detached eclipsing binaries, and water masers; (2) using these calibrated Cepheids to standardize SNIa observed with HST; and (3) measuring $H_0$ from SNIa distances in the Hubble flow, where cosmic expansion dominates.

Cepheid variables (Section 2.1.2) serve as primary standard candles due to their well-defined period–luminosity (P–L) relation, where the pulsation period correlates with intrinsic brightness. Systematic uncertainties, such as metallicity effects, crowding, and dust extinction, are mitigated through near-infrared photometry and consistent datasets from HST and JWST. The SHOES collaboration recently measured $H_0 =$
73.17 ± 0.86 km s⁻¹Mpc⁻¹using this approach, indicating a 5 – 6σ tension with Planck. JWST's independent observations, particularly in Cepheid-rich fields, have validated and refined these results by reducing crowding biases and confirming the reliability of the Cepheid calibration, strengthening the significance of the Hubble tension.

SNIa (Section 2.1.7) serves as the most common, far-field rung of the distance ladder, extending measurements into the Hubble flow. The Pantheon+ dataset provides precise $H_0$ measurements consistent with local results but in tension with early Universe estimates.

While the above tools remain central to the Hubble tension, offering the strongest leverage, alternative standard candles such as the TRGB, Mira variables, J-region Asymptotic Giant Branch (JAGB) stars, SNII, SBF, and the BTFR provide valuable cross-checks. These independent methods, while varying in calibration techniques and stellar populations, consistently yield higher values of $H_0$ than early Universe constraints from the CMB, emphasizing the robustness of the tension and fueling interest in theoretical refinements.

The TRGB method (Section 2.1.4) measures distances using the sharp luminosity cutoff where Red Giant Branch (RGB) stars ignite helium as a standard candle. Calibrated with galaxies like the Magellanic Clouds and NGC 4258, TRGB provides precise distance estimates. For state-of-the-art measurements, it yields consistent distance measures to SNIa hosts as Cepheids. TRGB remains a valuable cross-check for the Hubble tension, with minimal sensitivity to metallicity (if measured in the *I*-band) and dust.

Mira variables (Section 2.1.6) are long-period pulsating stars used as standard candles for measuring $H_0$. Their period–luminosity relation, especially in the near-infrared, provides reliable distances. Calibrated using galaxies like the LMC and NGC 4258, Miras offer an independent cross-check on $H_0$ with minimal sensitivity to metallicity and crowding.

JAGB stars (Section 2.1.8) are standard candles used for measuring $H_0$ based on their narrow luminosity range in the near-infrared. They are carbon-rich Asymptotic Giant Branch (AGB) stars in an advanced stellar phase, providing distance estimates at long range. Calibrated using galaxies like NGC 4258, JAGB measurements offer a route to calibrate SNIa. Though newer and less tested than Cepheids or TRGB, they contribute to cross-validation efforts.

SNII (Section 2.1.9) measure $H_0$ using the correlation between their luminosity and decline rate during the plateau phase. They offer an alternative to SNIa, aiding cross-checks in the Hubble tension.

The SBF method (Section 2.1.5) measures $H_0$ using pixel-to-pixel luminosity variations in elliptical galaxies. Calibrated with nearby galaxies, SBF provides $H_0$ estimates which are independent of SNIa.

The BTFR (Section 2.1.11) estimates $H_0$ using the correlation between a galaxy's rotational velocity and baryonic mass. Recent calibrations, particularly addressing zero-point uncertainties between different standard candles like Cepheids and TRGB, have refined the measurement to $H_0 = 76.3 \pm 2.1(\text{stat}) \pm 1.5(\text{sys}) \text{km s}^{-1}\text{Mpc}^{-1}$, yielding another SNIa-independent measurement and further emphasizing the tension with early Universe estimates.

As an alternative to the distance ladder measurements, masers offer independent insights into the Hubble constant $H_0$. The maser technique (Section 2.1.3) uses 22 GHz H₂O maser emissions from rotating disks around supermassive black holes to directly measure distances. Each of 5 masers (leaving NGC 4258 aside as its often used to calibrate the distance ladder) yields a geometric distance and $H_0 = 73.9 \pm 3.0 \text{ km s}^{-1}\text{Mpc}^{-1}$. The Megamaser Cosmology Project (MCP) has extended the precision of this method, producing results consistent with late-time measurements. Although rare, water masers provide a model-independent check, reinforcing the higher $H_0$ values obtained from local probes.

Strong gravitational lensing with time-delay measurements (Section 2.1.14) offers another independent method for estimating $H_0$. Multiple images of a background source, produced by a massive foreground lens, create time delays due to differences in the light paths.

These delays depend on the lensing geometry and the Universe's expansion rate. Collaborations such as H0LiCOW and TDCOSMO have applied this technique, yielding $H_0 \approx 74.2 \pm 2.6 \text{ km s}^{-1}\text{Mpc}^{-1}$, consistent with other late-time estimates.

Gravitational Wave (GW) events with electromagnetic counterparts, known as standard sirens (Section 2.1.17), provide an independent estimate of $H_0$ by measuring the luminosity distance from GWs emitted during compact object mergers. When an electromagnetic counterpart identifies the host galaxy, the redshift can be measured directly. Recent standard siren measurements yield $H_0 \approx 70.0 \pm 3.0 \text{ km s}^{-1}\text{Mpc}^{-1}$, offering a model-independent constraint on cosmic expansion and contributing to the Hubble tension debate. However, this approach is still severely limited by the rarity of GW events with EM counterparts, with only a single event which is too close for a reliable measurement of $H_0$.

CC (Section 2.1.13) estimate $H_0$ using the differential ages of passively evolving galaxies. By dating stellar populations, this method infers the expansion rate without relying on standard candles. However, unlike the use of distances, this measure is not empirical, depending on astrophysical modeling of aging stellar populations, their metallicities, star formation histories, and initial mass functions, with an uncertainty dominated by the modeling effort. Recent measurements suggest $H_0 \approx 70.6 \pm 6.7 \text{ km s}^{-1}\text{Mpc}^{-1}$, still consistent with both early and most late-time measurements. As a distance-ladder-independent probe, CC offers a different dimension of study of the Hubble tension.

HII galaxies (Section 2.1.10) measure $H_0$ using the correlation between their luminosity and emission line flux from ionized gas. The strong emission lines, primarily from young massive stars, allow for distance estimates based on the luminosity-size relation. Calibrated using nearby galaxies, HII galaxy measurements provide independent constraints on $H_0$ consistent with other local probes.

DESI has provided a novel avenue for measuring $H_0$ through the Fundamental Plane (FP) relation of early-type galaxies, with a calibration tied to the distance to the Coma cluster (Section 2.1.12). By leveraging DESI's extensive sample of over 4000 early-type galaxies, the FP relation was calibrated to provide precise distance estimates. For a distance to the Coma cluster of 98.5 ± 2.2 Mpc based on SNIa, DESI yields a local value of $H_0 = 76.5 \pm 2.2 \text{ km s}^{-1}\text{Mpc}^{-1}$. This result is in significant tension with the value inferred from Planck measurements where $\Lambda$CDM is assumed, and which predicts a distance to Coma exceeding 110 Mpc. Historically, the distance to the Coma cluster has ranged from 90-100 Mpc, highlighting the discrepancy. By extending the Hubble diagram with FP-calibrated distances, DESI highlights a persistent conflict between local measurements and early Universe predictions.

Extended QSO cosmologies (Section 2.1.15) constrain $H_0$ using correlations between QSO luminosity and variability timescales. Their ability to probe higher redshifts than standard candles makes them valuable for testing cosmic expansion over extended epochs. Similarly, GRB observations (see Section 2.1.16) use correlations between their luminosity and spectral properties to estimate $H_0$, serving as additional high-redshift probes relevant to the Hubble tension.

Despite significant methodological diversity and substantial precision improvements, the tension between early- and late-time measurements of the Hubble constant $H_0$ persists at a statistically significant level, exceeding $5.9\sigma$ (Planck 2018 vs. SH0ES, alone, but higher when combining measures). This discrepancy challenges the completeness of the standard $\Lambda$CDM model and raises the possibility of new physics.

Related to the $H_0$ tension there is the sound horizon tension, which refers to a discrepancy in the inferred comoving sound horizon scale at the end of the baryon drag epoch, $r_s^{\text{drag}}$, derived from early- and late-time cosmological measurements. This standard ruler, crucial for calibrating cosmological distances, is determined by the physics of the early Universe, specifically the acoustic oscillations in the photon–baryon plasma before recombination. It is primarily constrained by measurements of the CMB power spectrum, where Planck data assuming $\Lambda$CDM suggest $r_s^{\text{drag}} \approx 147.09 \pm 0.26 \text{ Mpc}$. However, late-time
<figure><img src="image_5.png" /><figcaption>Fig. 1. Summary of $H_0$ estimates from different cosmological probes with error bars smaller than 3.5 km s⁻¹Mpc⁻¹.</figcaption></figure>

measurements, such as BAO data combined with local distance ladder determinations of $H_0$, suggest a lower sound horizon, with BAO-based estimates yielding approximately 137 Mpc, a difference of about 7% and a tension of 2.6σ significance.

This discrepancy arises because $r_s^{\text{drag}}$ is directly connected to the expansion rate around recombination, meaning changes to the early Universe physics could shift its value. Solutions involving early-time modifications often require an enhanced expansion rate before recombination, which can reduce $r_s^{\text{drag}}$. Examples include models introducing additional relativistic species or EDE components. However, these scenarios face tight constraints from CMB observations, as the acoustic peak structure is highly sensitive to changes in photon diffusion and gravitational driving effects.

Late-time measurements rely on the BAO feature imprinted at the drag epoch, calibrated through local $H_0$ measurements, such as the SH0ES collaboration results. Since $r_s^{\text{drag}}$ anchors the BAO scale, discrepancies in $H_0$ estimates propagate into inferred sound horizon measurements. This tension therefore highlights the need for consistent cross-calibration between early- and late-time probes and motivates further investigation into both systematics and extensions to the standard model of cosmology.

The combination of the estimates resulting from these, and other, approaches to estimating the value of $H_0$ points to a significant tension in the value of this parameter. This tension in the expansion rate of the Universe at current times is further detailed in Section 2.1, with the most prominent estimates shown in Fig. 1 (see also Fig. 2) and possible solutions in Fig. 3.

Another interesting tension is that related to $S_8$, which highlights a persistent discrepancy between measurements of the amplitude of matter fluctuations on cosmological scales inferred from early and late Universe observations. The parameter $S_8$, defined as $S_8 = \sigma_8 \sqrt{\Omega_m/0.3}$, combines the clustering amplitude $\sigma_8$ (the root-mean-square of matter

fluctuations on scales of 8 h⁻¹ Mpc) with the present-day matter density parameter $\Omega_m$. It provides a key measure of the growth of cosmic structures, making it a crucial probe of the standard cosmological model.

Current measurements from early-time data, such as the Planck 2018 results, which constrain the CMB temperature and polarization anisotropies (Section 2.1.18.6), yield a high precision estimate of $S_8 = 0.834 \pm 0.016$ assuming the ΛCDM model. However, late-time measurements derived from weak gravitational lensing (Section 2.2.1) and galaxy clustering surveys (Section 2.2.2), including KiDS, DES, and HSC, consistently report lower values of $S_8$. For example, DES Year 3 results obtained from combined galaxy clustering and WL measurements suggest $S_8 = 0.772 \pm 0.017$, while KiDS-1000 and HSC report similarly low values around $S_8 \approx 0.76$ with comparable uncertainties. This tension, at the 2 – 3σ level, persists across multiple independent data sets, indicating a possible systematic deviation between early and late-time probes of structure formation.

The origin of this tension remains under debate. On the one hand, it could be driven by systematic uncertainties in the analysis of WL data. These systematics include shear calibration biases, uncertainties in photometric redshift estimates, and baryonic feedback effects that suppress the small-scale power spectrum due to processes like Active Galactic Nucleus (AGN) feedback and gas ejection from galaxies. DES and KiDS collaborations have both extensively explored the role of such systematics, yet the tension persists even after conservative scale cuts and improved modeling of non-linear clustering.

On the other hand, the $S_8$ tension might hint at the need for extensions to the standard ΛCDM model. One potential modification involves evolving DE models, where a time-dependent equation of state parameter $w(z)$ could alter structure growth rates. Another possibility is modifications to the theory of gravity itself, such as $f(R)$ models or scalar-tensor theories, which could modify the relationship between the
<figure><img src="image_6.png" /><figcaption>Fig. 2. Summary of $H_0$ estimates from all the cosmological probes in this White Paper following the order of the different sections.</figcaption></figure>

matter distribution and lensing potential. Additionally, the presence of massive neutrinos, which suppress structure formation at late-times due to their free-streaming behavior, could also contribute to lowering the observed $S_8$ value if the neutrino mass is larger than currently assumed in the base $\Lambda$CDM model.

Notably, cross-correlation measurements between WL and other probes, such as CMB lensing (e.g., from Planck, ACT, and SPT) and galaxy clustering from BAO surveys, have shown consistency with both early and late-time datasets, complicating the overall picture. Furthermore, analyses that vary only the normalization of the power spectrum, such as the $\sigma_8$ parameter itself, have not fully resolved the tension, suggesting a more complex interplay between cosmic structure formation and expansion history.

The tension in the $S_8$ parameter may be milder but there is a growing effort to improve the constraints in observational estimates of the amplitude of matter fluctuations. These are detailed in Section 2.2, with the most prominent estimates also shown in Fig. 4, and possible solutions in Fig. 5.

## 1.2. Early vs local measurements and efforts for a solution

The persistence of the $H_0$ tension across independent datasets suggests either unresolved systematic errors or the need for extensions to the $\Lambda$CDM framework. Numerous theoretical solutions have been proposed, aiming to reconcile the observed discrepancy while preserving the success of the standard cosmological model in describing early and late-time observables. The most prominent of these efforts for a solution are shown in Fig. 6, which are also highlighted below, and detailed in Section 4:

*   **Early Dark Energy (EDE):** EDE models (Section 4.1.1) propose a transient DE component that briefly dominates the cosmic energy budget before recombination, reducing the size of the sound horizon and allowing for a higher inferred $H_0$ from CMB data. This mechanism involves a scalar field with a potential that activates temporarily and dilutes before significantly affecting late-time cosmology. Several variants of EDE have been explored in the literature. Oscillatory EDE models, often motivated by axion-like fields, involve a scalar field oscillating around the recombination epoch, injecting energy into the expansion rate. NEDE (see Section 4.1.2) refines this idea by introducing a phase transition where the scalar field rapidly decays, creating a sharper and more controlled impact on the expansion rate. Adiabatic Fluctuation EDE models, meanwhile, adjust the evolution of the scalar potential to balance early- and late-time cosmological constraints, though these often introduce parameter degeneracies. While EDE models have shown promise in alleviating the $H_0$ tension by increasing the inferred value from early-time data, they also introduce challenges. The models can exacerbate tensions in other cosmological parameters, particularly the amplitude of matter fluctuations $S_8$ and the matter density $\Omega_m$. Depending on the specific data set, EDE can reduce the $H_0$ tension to $2-3\sigma$, however, recent CMB and LSS data sets show no evidence for this model and disfavor large fractions of EDE.

*   **Late Dark Energy (LDE):** Late Dark Energy (LDE) models (Section 4.2.1) attempt to address the Hubble tension by modifying the expansion history at low redshifts ($z \lesssim 1$). These models alter the equation of state parameter of DE, $w(z)$, away from the cosmological constant value $w = -1$. Some LDE scenarios involve a rapid transition in $w(z)$, where DE density evolves from a negative to a positive contribution, effectively accelerating the late-time expansion rate. Although LDE models can slightly raise $H_0$ without modifying the early Universe physics, they often face challenges in matching BAO and Type Ia supernova data simultaneously, as the shift in the expansion rate can introduce tension with large-scale structure constraints.

*   **Rapid transitions in the late-Universe:** Rapid transitions in the DE from negative to positive values in the late Universe (Section 4.2.2) exhibit a sign-switching action in the vacuum energy.
<figure><img src="image_7.png" /><figcaption>Fig. 3. Summary of models proposed to solve the $H_0$ tension in this White Paper following the order of the different sections.</figcaption></figure>

while leaving the DE magnitude unchanged. These models allow for the possibility of simultaneously increasing the value of $H_0$ while also suppressing changes in $S_8$. The transition point is established primarily by BAO data sets, while also being supported by other contributions. Most models express an abrupt transition with a discontinuity in the vacuum energy density.

* **Interacting Dark Matter and Dark Energy (IDE) Models:** Interacting Dark Matter and Interacting Dark Energy (IDE) models (Section 4.2.3) propose a non-trivial energy exchange between DM and DE, modifying both the cosmic expansion rate and structure growth. This interaction, described by a coupling function $Q$, allows energy to transfer from one component to the other. Depending on the direction and strength of the coupling, IDE can either slow down or accelerate the expansion rate. IDE models can increase $H_0$ estimates while modifying the growth of structure, offering a way to reduce the $S_8$ tension simultaneously. However, strong constraints from CMB lensing and BAO measurements limit the parameter space of IDE models, and they often require fine-tuning to remain consistent with multiple datasets.

* **Modified Gravity Theories:** MG models (Sections 4.3.1 and 4.3.2) propose extensions to GR by altering the Einstein-Hilbert action, introducing additional scalar degrees of freedom, altering the underlying geometry itself, or changing the spacetime dimensionality, among other alternatives, which can affect both the cosmic expansion rate and the growth of large-scale structure. These modifications can impact the lensing potential and structure formation, leading to changes in the inferred values of $H_0$ and $S_8$. Several frameworks have been explored to address the Hubble tension within MG. $f(R)$ gravity, for example, generalizes

GR by replacing the Ricci scalar with a nonlinear function of $R$, introducing an additional scalar mode that modifies both the background expansion and structure growth. Horndeski theories, which include non-minimally coupled scalar fields with derivative interactions, allow modifications to both the expansion history and lensing effects but often face constraints from CMB lensing and large-scale structure data. Other models, such as massive gravity and bimetric gravity, alter the graviton's properties, leading to modified cosmic acceleration and structure formation patterns. While some variants can reduce both the $H_0$ and $S_8$ tensions, they often require fine-tuning or introduce instabilities in the late Universe. Teleparallel gravity approaches, including $f(T)$ and $f(Q)$ theories, replace curvature with torsion or non-metricity as the primary geometrical property driving cosmic evolution. These models have been shown to affect the late-time expansion rate but remain under scrutiny for consistency with both early- and late-time datasets. Emerging curvature theories, such as AvERA [13,13,14] modify the late expansion history compared to the concordance model (Section 2.3.9) to solve the $H_0$ and ISW puzzles. While MG models offer a rich theoretical landscape to explore, many face difficulties in simultaneously resolving the $H_0$ and $S_8$ tensions while remaining consistent with solar system tests, CMB lensing, and BAO constraints.

* **Exotic Scenarios and Non-Standard Dark Matter Models:** Exotic scenarios (Sections 4.4.1, 4.4.2, and 4.4.3) explore extensions to the standard cosmological model involving non-standard physics in both the DM and DE sectors, which can influence the cosmic expansion history and structure formation. Among
<figure><img src="image_8.png" /><figcaption>Fig. 4. Summary of $S_8$ estimates from different cosmological probes.</figcaption></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 5. Summary of models proposed to solve the $S_8$ tension in this White Paper following the order of the different sections.</figcaption></figure>
these are decaying dark matter (DDM) models, where DM particles decay into lighter states. If this decay occurs before or during the recombination epoch, it can alter the expansion rate and reduce the sound horizon, leading to a higher inferred $H_0$. However, constraints from CMB lensing and large-scale structure limit the viability of DDM as a complete solution. Ultralight scalar fields, such as axion-like particles or fuzzy dark matter, have also been proposed. These fields can modify the standard expansion rate through their contributions to the cosmic energy density, potentially mimicking an additional relativistic species or altering structure formation on small scales. Nevertheless, precise measurements of the CMB power spectrum and the Lyman-alpha forest have placed stringent bounds on their properties. Primordial black holes and non-CDM models represent further exotic avenues, with the latter deviating from the standard cold, collisionless DM paradigm, often leading to a suppression of small-scale structure growth. While these models offer novel mechanisms to address cosmological tensions, most current models are tightly constrained by precision data from Planck, DESI, and the Pantheon+ supernova sample. However, they remain of theoretical interest, especially with upcoming data from CMB Stage-4 and JWST capable of probing extreme scenarios further.

* **Extra Relativistic Species and Neutrino Physics:** Extra relativistic species (Section 4.1.3) represent another class of proposed extensions aimed at resolving the Hubble tension. An increased number of relativistic degrees of freedom, commonly parameterized by $N_{\text{eff}}$, leads to a higher early Universe expansion rate and a reduced sound horizon, potentially increasing the inferred $H_0$. This scenario can arise from new light particles, such as sterile neutrinos or dark radiation. Sterile neutrinos, in particular, can act as an additional relativistic species if they decouple before standard neutrinos while self-interacting or secret neutrino interactions can modify the thermal history of the Universe and delay neutrino decoupling, effectively altering $N_{\text{eff}}$. However, current constraints from Planck, BAO, and CMB lensing data have limited viable deviations to $\Delta N_{\text{eff}} \approx 0.2$. While insufficient to fully resolve the Hubble tension, such scenarios remain relevant in the context of early Universe physics and potential beyond-the-Standard-Model extensions.

* **Local Void Hypothesis:** The local void hypothesis (Section 4.5.1) posits that the Milky Way galaxy (MW) resides in a large, underdense region of the Universe, potentially biasing measurements of the Hubble constant due to a locally faster expansion rate compared to the cosmic average. The idea suggests that a void could lead to a larger measured $H_0$ locally while being lower on cosmological scales. However, the size and depth of such a void required to explain the entire Hubble tension would be inconsistent with current observations. Studies using CosmicFlows-4 and Pantheon+ supernova data, as well as analyses of bulk flows, have not identified an underdensity significant enough to fully explain the observed discrepancy assuming $\Lambda$CDM.

* **Primordial Magnetic Fields (PMFs):** PMF (Section 4.5.2) contributions are generated before recombination could influence the early Universe's expansion history and the formation of large-scale structures. PMF contributions can modify the photon-baryon plasma dynamics, alter the acoustic peaks in the CMB power spectrum, and shift the inferred sound horizon, all of which can affect the measurement of $H_0$. However, current constraints from the CMB, including non-detections of Faraday rotation and the damping tail suppression, limit the strength of PMFs, making it unlikely that they can fully account for the observed tension.

* **Inflationary Models:** Certain inflationary scenarios (Section 4.5.3.2) propose modifications to the early Universe's physics that could indirectly affect the $H_0$ measurement. Models involving non-standard reheating phases, features in the inflationary potential, or a modified primordial power spectrum have

been investigated in this context. These scenarios can alter the early expansion history or the acoustic peak structure, impacting the inferred value of $H_0$ from CMB measurements. However, most inflationary modifications struggle to generate a significant enough shift in $H_0$ while remaining consistent with current CMB and large-scale structure data.

* **Varying Fundamental Constants:** The idea of varying fundamental constants (Section 4.8) explores the possibility that parameters such as the fine-structure constant $\alpha$, the electron mass $m_e$, or the proton-to-electron mass ratio $\mu$ could change over cosmic time or with spatial position. If fundamental constants were to evolve, they could impact key cosmological observables, such as the sound horizon and the physics of recombination, altering the inferred values of $H_0$. Scalar fields coupled to the electromagnetic sector, as in models like varying-$\alpha$ theories, could drive such changes. However, stringent constraints from CMB, BBN, and QSO absorption spectra have significantly limited the variation of some of these constants and their ability to resolve the Hubble tension, while the others face issues of finding a coherent theoretical framework that can explain their variation.

* **Local Physics Solutions:** Local physics solutions (Section 4.9) suggest that the $H_0$ tension could arise from new physical effects specific to the local Universe rather than requiring modifications to the global cosmological model. One possibility is a local modification to the gravitational constant, $G_{\text{eff}}$, which could alter the calibration of standard candles like Cepheid variables or SNIa, leading to a biased measurement of $H_0$. Another proposed local effect involves variations in the transparency of the Interstellar Medium (ISM) or modified extinction laws that could influence supernova observations. While some of these scenarios can explain a portion of the tension, they generally fail to provide a complete resolution and are often tightly constrained by measurements from cosmic bulk flows, galaxy clustering, and large-scale structure surveys.

* **Systematic Uncertainties and Calibration Issues:** A significant portion of the community continues to explore whether systematic errors could account for the observed tension. Potential sources include Cepheid calibration errors, host galaxy dust corrections, and selection effects in both early and late-time measurements. However, rigorous cross-calibrations between HST, JWST, and ground-based surveys like DESI and SHOES have yet to identify a significant bias capable of resolving the tension completely.

## 1.3. Additional curiosities and anomalies

Other cosmological anomalies beyond the primary $H_0$ and $S_8$ tensions have emerged in recent years, revealing potential challenges to the standard $\Lambda$CDM model. These discrepancies span a wide range of physical scales and cosmological probes, motivating deeper investigations into both systematic effects and new physics. Key anomalies include:

* **The $A_{\text{lens}}$ anomaly:** The $A_{\text{lens}}$ parameter, introduced as a phenomenological extension to the standard CMB analysis, quantifies the amplitude of lensing-induced smoothing of acoustic peaks in the CMB power spectrum. Planck data suggest a value for $A_{\text{lens}}$ slightly higher than expected under $\Lambda$CDM, with a significance of $2-3\sigma$, potentially pointing to unmodeled effects in CMB lensing or physics beyond standard cosmology (Section 2.3.1).

* **Evidence for nonzero curvature:** While the standard model assumes a flat universe, Planck's CMB data, when analyzed without external priors, have shown mild evidence for a closed geometry with $\Omega_k < 0$. However, this claim remains controversial, as BAO and lensing measurements are consistent with a flat universe within $1\sigma$ (Section 2.3.2).
<figure><img src="image_11.png" /><figcaption>Fig. 6. Fundamental physics provides a number of potential solutions to address the challenge of tensions in cosmology.</figcaption></figure>

* **CMB anisotropic anomalies:** Several statistically significant features have been identified in CMB data, including the hemispherical power asymmetry, alignments in low multipole moments, and the cold spot anomaly. While most of these could be attributed to cosmic variance, their persistence in multiple datasets (e.g., Planck, WMAP) raises the question of whether they hint at new physics or residual systematics (Section 2.3.3).

* **The $w_0w_a$ tension:** Discrepancies have emerged in constraints on the DE equation of state parameters, $w_0$ and $w_a$, particularly when comparing results from CMB data with late-time probes such as SN and DESI BAO. Some results suggest a deviation from the cosmological constant value of $w_0 = -1$, which could indicate evolving DE or unresolved systematics (Section 2.3.4).

* **Neutrino Tension:** The sum of neutrino masses, $\sum m_\nu$, inferred from cosmological data, such as CMB and BAO observations, has shown tension with the lower bounds set by terrestrial neutrino oscillation experiments, which require $\sum m_\nu \gtrsim 0.06$ eV (normal ordering, NO) and $\sum m_\nu \gtrsim 0.1$ eV (inverted ordering, IO). Recent DESI BAO data combined with Planck and ACT CMB constraints

yield tight upper limits, such as $\sum m_\nu < 0.05$ eV (2$\sigma$), challenging the inverted hierarchy and preferring the NO with a Bayes factor exceeding 46.5 in some datasets. Interestingly, some analyses allowing for extended models have reported a preference for *negative* neutrino masses, linked to enhanced CMB lensing signals in Planck and ACT data. This tension could stem from unaccounted systematics in lensing measurements or may indicate new physics beyond minimal $\Lambda$CDM, such as modified neutrino properties or evolving DE models (Section 2.3.5).

* **Cosmic dipole anomalies:** Tensions exist between the inferred cosmic dipole from the CMB and measurements from radio galaxies and QSOs. Separately, statistically significant directional variations in $H_0$ have been reported in both CMB and the local Universe, providing an alternative perspective on CMB anisotropic anomalies and bulk flow anomalies. These discrepancies could point to non-standard cosmic expansion or incomplete modeling of large-scale structure effects (Section 2.3.6).

* **Big Bang Nucleosynthesis (BBN) anomalies:** Discrepancies exist between deuterium and helium abundances inferred from BBN
and those predicted by CMB-based baryon density estimates, suggesting either measurement systematics or small deviations from standard physics during the early Universe (Section 2.3.7).

* **Lyman-α anomalies:** High-redshift Lyman-α forest measurements show mild tension with low-redshift constraints on the matter power spectrum, raising questions about the evolution of structure and thermal history of the Intergalactic Medium (IGM) (Section 2.3.8).

* **Integrated Sachs–Wolfe (ISW) and cosmic superstructures:** Measurements of the ISW effect, which probes late-time structure growth through correlations between CMB and large-scale structure data, have shown excess correlations on large scales, potentially linked to cosmic superstructures (Section 2.3.9).

* **Cosmic void anomalies:** Observations of cosmic voids have revealed unexpected properties in their size distribution and their contribution to the ISW effect, potentially suggesting deviations from the standard growth of structure (Section 2.3.11).

* **Fast Radio Burst (FRB) probes:** FRB observations have recently emerged as a powerful probe of cosmic tensions, offering a unique way to constrain the expansion history and large-scale bulk flows. Preliminary results suggest some inconsistencies with early universe predictions. It is crucial to emphasize that in estimating $H_0$ from FRBs, a prior assumption of the underlying cosmological model is required. Specifically, the $H_0$ values presented in the associated Whisker plot are computed under the assumption of a $\Lambda$CDM cosmology. Deviations from the standard $\Lambda$CDM framework are expected to yield different $H_0$ estimates, reflecting the model dependence inherent in such measurements. Furthermore, the constraints on $H_0$ have progressively tightened over time, primarily due to the increasing number of well-localized FRBs and improvements in the statistical methodologies applied to their analysis (Section 2.3.12).

* **Radio background excess:** Several observations of the diffuse radio sky have reported an unexplained excess in both the surface brightness and the anisotropy power of the radio background, which could be linked to new diffuse or low flux radio sources, PMFs, exotic decays, or residual systematics across a range of radio observations (Section 2.3.13).

* **Bulk flow anomalies:** Peculiar velocity measurements and bulk flows inferred from galaxy catalogs show a higher amplitude than predicted by $\Lambda$CDM, potentially pointing to the presence of large-scale structures beyond the standard model (Section 2.3.14).

* **Ultra Long Period Cepheids:** These variable stars have been proposed as potential standard candles, but their properties appear inconsistent with standard stellar evolution models, raising questions about distance ladder calibrations (Section 2.3.15).

Ongoing and upcoming surveys such as Euclid, The Rubin Observatory's Legacy Survey of Space and Time (LSST), and the Roman Space Telescope will play a pivotal role in clarifying these anomalies. Additionally, methodological advances, including cross-correlations between multiple probes and the application of Machine Learning (ML) techniques (Sections 3, 3.2–3.5), will help assess whether these tensions are genuine physical discoveries or the result of residual systematics. A deeper understanding of these anomalies is essential to determine whether extensions to the standard $\Lambda$CDM framework are required and to uncover the fundamental nature of DE, DM, and cosmic structure formation.

## 1.4. Data analysis: How to tackle the problem

The analysis of cosmological data has seen remarkable advancements, leveraging a wide range of statistical, computational, and interdisciplinary approaches. These methodologies have not only refined parameter constraints but also opened new avenues to tackle cosmological tensions such as the $H_0$ and $S_8$ discrepancies. These tensions challenge

the completeness of the $\Lambda$CDM model, suggesting the possibility of new physics or unresolved systematics.

Cosmology simulators and MCMC techniques (Section 3.1) are foundational tools for exploring high-dimensional parameter spaces. They allow for robust testing of cosmological models against observational data while accounting for degeneracies and systematics. Advances in MCMC methods, such as nested sampling and Hamiltonian Monte Carlo, have improved the efficiency and accuracy of parameter estimation, particularly when combining data from the CMB, BAO, and SN datasets.

ML approaches (Section 3.2) have emerged as powerful tools in cosmological analysis. Neural networks, GP, and decision trees have demonstrated their utility in extracting patterns, accelerating data analysis, and refining model predictions. For example, ML techniques have been used to process large-scale structure data and improve constraints on DE models. Hybrid approaches, combining ML with traditional statistical methods, have further enhanced the reliability of these analyses by integrating the strengths of both paradigms.

Reconstruction methods (Section 3.3) have also played a critical role, enabling the inference of the Universe's expansion history from observational data. Techniques such as GP regression have been applied to reconstruct the Hubble parameter $H(z)$ and the growth of cosmic structures without assuming a specific cosmological model. These methods reduce systematic biases and provide model-independent insights into cosmological tensions.

Bio-inspired algorithms, particularly GA (Section 3.4), have introduced novel ways to optimize model selection and parameter estimation in cosmology. These algorithms, inspired by evolutionary processes in nature, use mechanisms such as selection, mutation, and crossover to identify optimal solutions in complex and high-dimensional parameter spaces. GAs are particularly well-suited for exploring cosmological models where traditional gradient-based methods may struggle, such as those involving nonlinear dynamics or multimodal likelihood functions. By iteratively refining a population of candidate solutions, GAs can efficiently locate regions of interest in the parameter space, even in the presence of degeneracies or non-Gaussian distributions. Applications of GAs in cosmology have been diverse. They have been employed to address key tensions, such as the $H_0$ and $S_8$ discrepancies, by testing alternative models including EDE, Interacting Dark Matter (IDM)-DE scenarios, and MG theories. For instance, GAs have been used to explore the parameter space of IDE models, identifying regions that minimize tensions with both early- and late-time observations. Their flexibility allows for the incorporation of priors from ML or other inference techniques, further enhancing their robustness. Moreover, hybrid approaches that combine GAs with ML have shown promise in improving the convergence and accuracy of solutions. These methods leverage the pattern recognition capabilities of neural networks or decision trees to guide the evolutionary search process, thereby reducing computational overhead while maintaining accuracy. In addition to model testing, GAs have proven effective in enhancing observational strategies. By simulating survey configurations and optimizing the allocation of observational resources, they help maximize the scientific output of upcoming missions. This includes optimizing survey strategies for WL, galaxy clustering, and other probes of large-scale structure. The adaptability of GAs also extends to their role in reconstructing the initial conditions of the Universe from observed data. By evolving populations of initial conditions, these algorithms can identify those that best reproduce observed structures, offering insights into the fundamental physics of the early Universe. Overall, bio-inspired algorithms like GAs represent a powerful addition to the cosmological toolkit, enabling the exploration of complex models and the refinement of observational strategies. Their continued integration with other advanced methods, such as ML and high-performance simulations, holds great potential for addressing the most pressing tensions in cosmology and uncovering new physics.
Cosmological simulations (Section 3.5) have become indispensable for interpreting observational data and testing theoretical models. High-resolution *N*-body simulations, hydrodynamic models, and semi-analytic methods provide a detailed understanding of the large-scale structure formation and the interplay of DM and DE. These simulations enable the calibration of observables such as WL signals, galaxy clustering, and BAO measurements, thereby improving the accuracy of cosmological parameter constraints.

Statistical tools, including profile likelihoods, have gained prominence in cosmology for estimating confidence intervals and assessing parameter significance. Unlike traditional Bayesian methods, profile likelihoods provide a frequentist alternative (Section 3.6) that is less sensitive to prior assumptions. This makes them particularly useful in scenarios where prior information is limited or where strong assumptions may bias the results. Profile likelihoods excel in analyzing models with complex parameter spaces, including those with non-Gaussian distributions or significant degeneracies between parameters. Such complexities often arise in cosmological contexts, for instance, when exploring EDE models, IDE scenarios, or the inclusion of additional relativistic species. These models frequently introduce additional degrees of freedom, leading to correlations that can obscure parameter constraints when traditional Bayesian approaches are employed. A key advantage of the profile likelihood approach is its ability to disentangle degeneracies and provide robust confidence intervals without relying on the full posterior distribution. This is achieved by profiling the nuisance parameters, effectively marginalizing them without requiring explicit integration. As a result, the profile likelihood method can highlight the true parameter dependencies and offer more transparent interpretations of the data. Applications of profile likelihoods in cosmology have included constraining the equation of state of DE, testing deviations from the standard ΛCDM model, and evaluating the statistical significance of cosmological tensions, such as those involving $H_0$ and $S_8$. For example, in the context of EDE, profile likelihoods have been used to assess whether parameter shifts can resolve these tensions or whether they indicate systematic effects in the data. The computational efficiency of profile likelihoods also makes them a valuable tool in high-dimensional analyses, particularly for next-generation cosmological surveys. Their ability to isolate relevant subspaces of parameter space enables efficient exploration of hypotheses while minimizing the computational cost associated with evaluating full posterior distributions. Furthermore, recent advances in computational methods have improved the applicability of profile likelihoods. Techniques such as adaptive mesh refinement and ML-assisted likelihood evaluations have reduced the computational demands of high-precision analyses. These innovations make profile likelihoods increasingly attractive for analyzing the large datasets expected from upcoming missions like the Simons Observatory, Euclid, and the Roman Space Telescope.

These methodologies, often used in combination, reflect a collaborative effort across theoretical, computational, and observational domains. For example, the integration of MCMC methods with ML-enhanced simulations and reconstruction techniques has provided a comprehensive framework for addressing persistent cosmological tensions. This synergy has also enabled the exploration of exotic scenarios, such as decaying DM, PMFs, and varying fundamental constants, which aim to explain anomalies in the current cosmological framework.

The ongoing advancements in data analysis tools and methodologies are pivotal for the next generation of cosmological surveys, including the Simons Observatory, CMB—Stage 4 (CMB-S4), Euclid, and the Roman Space Telescope. These surveys will generate unprecedented datasets that require state-of-the-art techniques to extract meaningful insights. A schematic is shown in Fig. 7 of these methods. By embracing these innovations, the cosmological community continues to push the boundaries of our understanding of the Universe, addressing existing tensions and paving the way for new discoveries in fundamental physics.

## 2. Observational cosmology and systematics

### 2.1. $H_0$ tension: Measurements and systematics

**Introduction:** Adam Riess

It is cliché to say we live in the era of precision cosmology — but it is true. Over the past several decades, the type, scope, and precision of cosmological measurements have grown enormously. Two of the most powerful tools, SNIa, and BAOs, joined the cadre of first-rank indicators during this time. Other observables — such as the CMB, gravitational lensing, primordial abundances, and components of the distance ladder — have been refined and matured, sharpening our view of the Universe. It is an exhilarating time to be a cosmologist. The surprising and non-intuitive composition of the Universe demands the full use of our observational toolkit. Whether the correct cosmological model is the simplest form of ΛCDM or one with additional complexities will ultimately be determined by the quality of our measurements. In the following sections, you will read how the astronomical zoo of objects and features has been employed to produce precision tests of the Universe, with a level of control over systematic errors once known only to particle physics. To be fair, not every tool and technique has reached the same level of robustness (and some may never do so), but this is something you can judge for yourself as you explore the state of each art. Perhaps the biggest question for observational cosmology in the 2020s is what to make of the growing evidence of tensions. The first appearance of any tension or anomaly is usually attributed to experimental error or systematics — an assumption that makes sense when playing the odds. However, signals that may herald new physics will be treated the same unless critical thinking and extensive analysis follow. At present, one or more tensions have surpassed thresholds of statistical significance, reproducibility, and independent cross-checks, earning them the continued attention of the field. In the end, as Einstein once said, the Universe (or Lord) is subtle but not malicious. Our goal is to measure these subtleties.

#### 2.1.1. *The distance ladder*

**Coordinator:** Louise Breuval

**Contributors:** Adam Riess, Giulia De Somma, Leandros Perivolaropoulos, Lluís Galbany, Lucas Macri, Richard I. Anderson, Siyang Li, and Vladas Vansevicius

The Hubble constant measures the present expansion rate of the Universe. This cosmological parameter represents the slope of the redshift-distance relation, $cz = H_0D$, in the limit of $z \sim 0$. Only distant galaxies are sensitive to the Universe's expansion, unlike nearby systems which are dominated by local gravitational interactions. The best method to reach galaxies in this regime, called the *Hubble Flow* ($0.02 < z < 0.20$), is to build a distance ladder based on a succession of distance indicators. The most widely used and best calibrated distance ladder is based on three rungs. First, geometric distances are used to calibrate the period–luminosity relation of Cepheid variables in nearby galaxies. This law is, in turn, adopted to calibrate SNIa on the second rung, which is limited to the volume where both Cepheids and SNIa are observable. Finally, on the third rung, distances and redshifts of SNIa in the *Hubble Flow* directly measure $H_0$. Although Cepheids provide the most homogeneous and reliable set of distances to many nearby galaxies, they can be substituted by alternative standard candles such as TRGB stars [15,16], JAGB [17,18], or Mira variables [19,20]. On the second rung, SNIa can be replaced by SBF [21,22], SNII [23,24] or the Tully–Fisher relation [25]. These methods provide valuable independent checks of the distance ladder.

A Cepheid-SNIa distance ladder was included in the HST Key Project on the Extragalactic Distance Scale [27] result of $H_0 = 72 \pm 8 \text{ km s}^{-1} \text{Mpc}^{-1}$ (which calibrated multiple secondary distance indicators using Cepheids). Since then, this measurement was significantly
<figure><img src="image_12.png" /><figcaption>Fig. 7. Schematic of the different data analysis approaches used in the cosmology community.</figcaption></figure>

improved using state-of-the-art data from later HST instruments, homogeneous calibration, and near-IR Cepheid observations (Fig. 8). The first rung of the distance ladder is currently supported by four geometric calibrators, each providing a direct absolute determination of the Cepheid luminosity: Gaia DR3 parallaxes of Cepheids and host clusters in the MW [28], detached eclipsing binaries distances in the Large and Small Magellanic Clouds (SMC) [29,30], and the water maser distance to NGC 4258 [31]. The second rung now comprises a total of 42 SNIa in 37 galaxies, where all hosts are observed with the same instrument (Wide Field Camera 3 (WFC3)) on the same HST, an investment of more than 1000 orbits of observing time. On the last rung, the Pantheon+ team provides ~ 300 SNIa in the Hubble Flow with the highest quality data and calibration, standardized across many surveys [32,33]. These recent developments provided the most precise $H_0$ measurement from a simultaneous fit of the three rungs with $73.17 \pm 0.86 \text{ km s}^{-1} \text{Mpc}^{-1}$ [34,35]. This HST-based measurement has

recently been confirmed with observations of a subsample of SNIa host galaxies with JWST, which show excellent consistency between both telescopes [26] (Fig. 9). Regardless of the method, essential elements for a precise $H_0$ measurement in the late Universe include the use of near-infrared photometry to minimize the impact of dust [36] and consistent data to cancel flux calibration errors between rungs.

### 2.1.2. Cepheid variables as standard candles

**Coordinator:** Louise Breuval

**Contributors:** Adam Riess, Giulia De Somma, Leandros Perivolaropoulos, Lluís Galbany, Lucas Macri, Richard I. Anderson, Siyang Li, and Vladas Vansevicius

#### 2.1.2.1. Cepheid variables as standard candles
Classical Cepheids are luminous evolved stars located in the instability strip region of the
<figure><img src="image_13.png" /><figcaption>**Fig. 8.** The Cepheid-SNIa distance ladder [26]. The left-hand panel (first rung) shows the four anchor galaxies with geometric distances which are adopted to calibrate Cepheids. In the middle (second rung), 42 SNIa are calibrated with Cepheids. On the right-hand panel (third rung), redshifts and distances of SNIa in the Hubble Flow directly measure the expansion rate of the Universe.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 9.** Comparisons of $H_0$ between HST Cepheids (baseline, green circles) and JWST Cepheids (squares) for different SNIa host subsamples. The CCHP (blue) and SHOES (red) subsamples selected for JWST observations produce a difference of $3-4 \text{ km s}^{-1} \text{Mpc}^{-1}$ in $H_0$ owing to selection. The HST and JWST distance measurements themselves are in good agreement [26]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Fig. 10.** Multi-band light curves for Cepheid OGLE-SMC-CEP-4697 based on a compilation of data from Refs. [37–39].</figcaption></figure>

Hertzsprung–Russell diagram. These stars undergo regular radial pulsations driven by two mechanisms: the $\kappa$ (opacity) and $\gamma$ (adiabatic exponent, $\Gamma_3 - 1$) mechanisms, which operate in the partial ionization zones of abundant elements such as H, He, and He+ [40,41]. Cepheids are regarded as the “gold standard” for distance measurements due to their tight period–luminosity (P–L) relation [42], expressed as $M = a \log P + b$, where M is the absolute magnitude, P is the pulsation period, and a and b are constants. By measuring a Cepheid’s P, its M can be determined, and its distance D (in Mpc) can be inferred using the distance modulus: $m - M = 5 \log_{10}(D) + 25$, where m is the apparent magnitude. From a theoretical perspective, the P–L relation arises from fundamental principles of stellar astrophysics [43]. The pulsation period is related to the mean density $\rho$ of the star by $P \propto \rho^{-1/2}$, where $\rho \propto M/R^3$ ($M$ and $R$ are the stellar mass and radius, respectively). The Stefan–Boltzmann law, $L \propto R^2 T_{\text{eff}}^4$, and the mass–luminosity relation, $L \propto M^{3.5}$, link the star’s luminosity ($L$) and effective temperature ($T_{\text{eff}}$) to its pulsation period, providing a direct foundation for the P–L relation.

Cepheids play a key role in the distance ladder [34]. Their P–L relation is locally calibrated with geometric distances and is used, in turn, to calibrate SNIa on the second rung. Cepheids are easy to identify due to their periodicity and are bright enough to be observed up to about 50 Mpc with the current generation of space telescopes. They are generally identified at optical wavelengths [44], where their pulsation amplitudes peak (~1 mag in the V band), allowing precise determination of their pulsation periods (Fig. 10). Follow-up observations in the near-infrared minimize the impact of interstellar dust on their magnitudes. P–L relations are typically constructed using intensity-averaged magnitudes over the pulsation cycle. These averages are derived either through extensive photometric sampling or by fitting random-phase measurements with template light curves [45,46]. Using mean magnitudes significantly reduces the P–L dispersion by a factor of ~2 compared to random-phase measurements.

However, systematic uncertainties in Cepheid distances may arise due to differences between nearby calibrators and distant Cepheids observed in SNIa host galaxies. These include chemical composition, period range, crowding, dust properties, and potential nonlinearity of the P–L relation (see Ref. [47] for a comprehensive review). In the following sections, we outline the most significant systematics affecting Cepheid distances, and we propose solutions to mitigate them.

#### 2.1.2.2. Improvements in reducing the systematics.

**Photometric systems** Combining multiple photometric systems to observe Cepheids in the first and second rung introduces a 1.4–1.8% systematic error in distance measurements, as described in Ref. [48]. To mitigate this, it is advantageous to use the same telescope and instrument, thereby minimizing flux-calibration errors. Ground-based telescopes, while offering dedicated facilities with high availability [38], are impacted by atmospheric absorption and scattering. In contrast, space-based observatories like the HST and JWST provide superior image quality, are unaffected by weather, exhibit excellent long-term stability, and are not constrained by the day-night cycle. Prior to the launch of JWST, WFC3 on HST was the only instrument capable of resolving Cepheids in the most distant SNIa host galaxies at optical and near-infrared wavelengths [34]. While WFC3 is well-suited for observing both faint and bright targets, it suffers from a slight non-linearity in the infrared. Specifically, photons from faint stars (low count rates) are measured less efficiently than those from bright stars (high count rates), introducing a Count Rate Non-Linearity (CRNL). Though small, this bias is critical for achieving percent-level precision and amounts to $0.0077 \pm 0.0006$ mag/dex [49]. A CRNL correction is typically applied to bright Cepheids in anchor galaxies to ensure consistent calibration [34,35].

**Crowding** Separating Cepheids from their surrounding stellar populations is one of the most significant challenges in measuring Cepheids in SNIa host galaxies [15]. Contamination from redder stars, such as RGB and AGB stars along the line of sight, limits the precision of Cepheid measurements, especially in the near-infrared. A common approach to address crowding is to add artificial stars of known brightness at random positions near Cepheids and recover their magnitudes using the same photometric methods applied to real stars [44]. This allows for a statistical correction for crowding (although such corrections remove the bias, but not the added scatter or crowding noise). With the advent of JWST, the ability to separate Cepheids from background stars has improved significantly, resulting in a substantial reduction in crowding noise. Recent JWST observations of Cepheids in SNIa host galaxies including the anchor galaxy NGC 4258 [50,51] have demonstrated excellent agreement with prior HST measurements, achieving a mean difference of only 0.01 mag. Furthermore, JWST data reduced the scatter in the P–L relation by a factor of 2.5 (Fig. 11). This result decisively rules out Cepheid crowding from HST photometry as the cause of the Hubble tension, with a $8.2\sigma$ CL.

**Metallicity** The luminosity of Cepheids at a given period is known to correlate with their chemical composition, but the sign and magnitude of this dependence ($\gamma$) have historically been difficult to constrain [52–54]. Accounting for the metallicity dependence is essential in the distance ladder, particularly when calibrating Cepheids in the Magellanic Clouds, which are more metal-poor than typical SNIa host galaxies [35,48]. However, Cepheids in NGC 4258 and the MW resemble those in large spiral SNIa hosts, making metallicity differences negligible for determining $H_0$. Recent calibrations of the metallicity effect, leveraging accurate distances and expanded Cepheid samples, have converged on a consensus. These calibrations include studies using Gaia parallaxes and individual spectroscopic abundances of MW Cepheids [55–57] and combined analyses of MW and Magellanic Cloud Cepheids with geometric distances [35,58]. The derived metallicity dependence ($\gamma$) lies between –0.2 and –0.3 mag/dex, with the negative sign indicating that metal-rich Cepheids are intrinsically brighter than metal-poor ones. Recent theoretical studies have corroborated both the sign and magnitude of this effect [59,60], particularly for the Wesenheit index (a dereddened magnitude combining multiple filters [61]) employed in the SH0ES distance ladder [34]. For nearby galaxies, direct [Fe/H] abundances can be determined from high-resolution spectroscopy of Cepheids [56,62]. In more distant galaxies, metallicities are inferred from the [O/H] gradient measured via HII region spectroscopy [63–65] or optical Integral Field Spectroscopy (IFS) of calibrator galaxies [66]. These methods are consistent in the
<figure><img src="image_16.png" /><figcaption>**Fig. 11.** Cepheid P–L relations in the Wesenheit index obtained from HST/F160W (gray) and JWST/F150W (red), showing the 2.5 reduction in scatter in NGC 4258 (left) and NGC 5584 (right). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

*Source:* Figure taken from Ref. [51].

MW, where Cepheid spectroscopy agrees with HII region gradients to within $1\sigma \sim 0.05$ dex [34]. Because the metallicities of Cepheids in SNIa host galaxies closely match those in the anchor galaxies, the Hubble constant is only weakly affected by the metallicity effect. Neglecting the correction would change $H_0$ by just $0.5 \text{ km s}^{-1} \text{Mpc}^{-1}$ (in the higher $H_0$ direction) [34]. Although the impact on $H_0$ is minimal, the metallicity correction plays a crucial role in ensuring consistency among anchor galaxies while independently matching the radial dependence with Gaia parallaxes in the MW.

**Binaries** While crowding can be mitigated using artificial star measurements, unresolved companion stars physically associated with Cepheids can bias their measured flux, thereby affecting inferred distances. A significant fraction of Cepheids reside in binary or multiple systems [67], and unresolved companions typically increase the measured brightness of Cepheids. However, because unresolved binarity is equally present in both SNIa hosts and anchor galaxies, its effect largely cancels out in calibrated distances. In the Wesenheit index adopted in the SH0ES distance ladder, [68] used synthetic populations of binary Cepheids to estimate the impact of binarity, finding it to be a minor effect, contributing only 0.004 mag in distance modulus, or a 0.1% change in $H_0$. Similarly, [69] determined that the primary source of flux contamination arises from Cepheids located in open clusters, leading to an overestimate of $H_0$ by approximately 0.23%. However, the fraction of Cepheids in open clusters is relatively small (see also Refs. [46,70]). Thus, while binarity and clustering introduce minor biases, their overall impact on $H_0$ remains negligible at the current level of precision.

### 2.1.2.3. Perspectives.

**Dust laws** Dust has long been a critical systematic uncertainty in the distance ladder [71]. To correct for dust, one can either subtract $R_\lambda \times E(V-I)$ to Cepheid apparent magnitudes or adopt a reddening-free “Wesenheit” magnitude (see Ref. [34,61] for details). Additionally, to the uncertainties in the reddening $E(V-I)$ itself, the value of the $R_V$ parameter is not independently known in each SNIa host galaxy and at each Cepheid position, but rather is assumed to match the MW [34,72]. However, the use of near-infrared observations greatly limits the impact of reddening, reducing the size of $R_\lambda$ to $\sim 0.4$ and variations due to these different values of $R_V$ to $\pm 0.03$. Varying the reddening law across different hosts first requires subtracting the intrinsic color of Cepheids in order to separate the component of the color that results from dust reddening (see Sect. 6.3 and Appendix D in Ref. [34]). A complete sampling of the reddening curve at long and short wavelengths in each host as well as an improved characterization of the dependence of $R_V$ with host properties such as mass, star formation rate, and color [73] can further reduce this source of uncertainty in Cepheid measurements.

**Cepheid parallaxes** Early measurements of trigonometric parallaxes were obtained with the Fine Guidance Sensor (FGS) on HST [74] and later by spatial scanning with HST/WFC3 [75,76] for $\sim 10$ MW Cepheids. The European Space Agency (ESA) Gaia mission now provides parallaxes for thousands of Cepheids with individual precision at the $\sim 5\%$ level [28]. The main systematic uncertainty in this method

arises from the parallax zero-point, a small corrective term estimated with distant QSOs, LMC stars and binaries. The Early Data Release 3 constrained this term and characterized its dependence on magnitude, color, and position [77]. However, MW Cepheids are bright, resulting in limited sampling of the offset term for these stars. Fortunately, the parallax zero-point offset is an additive term, while the distance scale derived from parallaxes is multiplicative. This distinction allows for separation of the two terms, provided a sufficient range of Cepheid distances is available. The analysis of 75 MW Cepheids with high S/N and HST/WFC3 photometry in Ref. [78] yielded a residual parallax offset of $-14\mu$ as for Cepheid-like bright stars, which aligns well with the majority of independent studies for stars in this magnitude range [79]. MW Cepheids thus remain the strongest anchor of the distance ladder.

Greater individual precision can be achieved with Cepheids in open clusters by using the average parallax of cluster members [80,81]. Cluster members, being fainter than Cepheids, allow for a more accurate calibration of the parallax offset. In this magnitude range, studies by Refs. [82,83] find a zero-point consistent with zero, demonstrating good agreement between P–L relations derived from individual Cepheids and cluster members. These results further confirm the robust accuracy of Gaia parallaxes across a broad range of magnitudes.

**New anchors for the cepheid distance ladder** The distance ladder is currently supported by geometric distances in four galaxies: Gaia parallaxes in the MW [78,82], detached eclipsing binary (DEB) distances in the Large [29,48] and Small [30,35] Magellanic Clouds, and the water maser distance to NGC 4258 [31,84]. Adding new anchors to the Cepheid calibration would reduce uncertainties and strengthen the first rung of the distance ladder. A galaxy can serve as an anchor if it meets two conditions: (1) having Cepheid photometry measured in the same system (e.g., HST/WFC3) as the rest of the distance ladder and (2) having a sufficiently precise geometric distance. The two nearby galaxies, M31 and M33, are excellent candidates for this role, as they contain large Cepheid samples observed with HST/WFC3 [46,85]. However, they remain beyond the reach of late-type DEBs, which are too faint at these distances or suffer from imprecise geometric distances obtained with other methods [86]. On the other hand, early-type DEBs are brighter and could enable geometric distance measurements for M31 and M33 [87]. This method relies on model atmosphere theory, in contrast to the 1% precision surface brightness color relation used for late-type DEBs in the Magellanic Clouds. Significant advancements in these methods are anticipated [88,89], and they should soon provide additional robust anchors for the Cepheid distance ladder.

**Future prospects** Ongoing surveys (e.g., Zwicky Transient Facility (ZTF)) and the next generation of ground-based (Rubin, Extremely Large Telescope (ELT)) and space telescopes (Roman) will provide new insights into the Cepheid distance ladder [90,91]. These advancements will enable the discovery of new Cepheids through deep optical time-series observations of distant galaxies, extending the volume of the local Universe where Cepheids can be studied. High-precision photometry for bright stars in the MW and Magellanic Clouds will be measured by the PhotSat mission in optical bandpasses, which will also help
characterize the extinction law in different environments. Direct abundance measurements obtained with ground-based ELT will complement current datasets and extend the range of distances over which different metallicity tracers — such as Cepheids, blue supergiants, or HII regions — can be compared [62,63,92,93]. Expanding Cepheid samples in both nearby and distant galaxies will allow further tests of the linearity and universality of the P–L relation, a subject of ongoing debate [94–96]. In SNIa host galaxies, long-period Cepheids dominate as they are the brightest, while in nearby anchors, they may saturate or have invalid parallaxes. Conversely, P–L relations in nearby galaxies often include mostly short-period Cepheids. This period disparity could influence the inferred distances in the event of a P–L break. Recent studies [34,97,98] show no evidence for a non-linear P–L relation and demonstrate that allowing for different slopes does not improve the P–L fit. [34] also finds the P–L slope consistent with a single value within $1\sigma$ across the current sample of host galaxies. The significance of this effect will be confirmed with larger samples at greater distances. Additionally, the metallicity dependence of the P–L slope [55,59] will be clarified through observations in different environments.

### 2.1.3. Maser driven constraints

**Coordinator:** Cheng-Yu Kuo

**Contributors:** Dom Pesce, Jim Braatz, Mark Reid, and Violetta Impellizzeri

For the purpose of measuring $H_0$ using accurate distances external galaxies, the so-called H$_2$O maser technique, e.g., see Ref. [99, 100] provides a novel tool that allows one to by-pass the traditional extragalactic distance ladders and measure the angular-diameter distance to a galaxy in a single step without relying on the CMB. This method involves sub-milliarcsecond resolution imaging and single-dish monitoring of the 22 GHz H$_2$O maser emission from sub-parsec circumnuclear disks at the center of active galaxies. These disk maser emissions, which arise from the $J_{K_{-}K_{+}} = 6_{16} - 5_{23}$ transition<fnref n="1" /> of the ortho-H$_2$O molecule, usually have extremely high surface brightnesses, permitting mapping with Very Long Baseline interferometry (VLBI) that allows for a unique probe of the gas distribution and kinematics on sub-parsec scales at the centers of distant galaxies [102].

As revealed in the prototypical maser galaxy NGC 4258, e.g., see Ref. [103,105], the masing gas in a disk maser system often resides in a nearly edge-on thin disk around the central supermassive black hole (BH). The rotation curve of the gas can be nearly perfectly traced by Keplerian rotation, enabling black hole (BH) mass measurements to percent-level accuracy, e.g., see Ref. [106,107]. Given the assumption that the gas dynamics is dominated by the gravity of the central BH, the simplicity of disk geometry and kinematics allow one to use the orbital radii $r_{sys} = v_{sys}^2/a_{sys}$ of the systemic maser components, see Fig. 12, e.g., see Ref. [106], as a standard ruler for measuring the angular-diameter distance $D_A$ of the galaxy, e.g., see Ref. [105], where $v_{sys}$ and $a_{sys}$ standing for the orbital velocity and centripetal acceleration of a systemic maser component. By using $r_{sys}$ as the standard ruler, one can easily show that the angular-diameter distance of a maser galaxy can be expressed as $D_A = v_0^2 \sin(i)/a\Delta\theta$, where i and $\Delta\theta$ indicate the maser disk inclination and the angular radius of the systemic masers, respectively. As long as the masing components of the gas follow circular orbits, one can determine an accurate galaxy distance by measuring the four disk parameters including $a_{sys}$, $v_{sys}$, i, and $\Delta\theta$, where $a_0$ can be measured from single-dish monitoring of the maser lines, e.g., see Ref. [100] and the rest of the three parameters can be obtained by modeling the maser disk in three dimensions, e.g., see Ref. [108,109].

To apply this maser technique to galaxies in the local Universe, one has to first search for maser disks similar to NGC 4258 that

<figure><img src="image_17.png" /><figcaption>**Fig. 12.** The distribution and Keplerian rotation curve [103,104] of the 22 GHz H$_2$O masers in NGC 4258. The red, blue, and green dots in the plot represent the redshifted, blueshifted, and systemic maser components, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

allow for robust disk modeling. To this end, the Megamaser Cosmology Project (MCP; [111,112]) has carried out an extensive survey of H$_2$O megamaser emission from > 4800 AGNs [113,114] within redshift $z \le 0.05$, resulting in the detection of $\ge 30$ candidate disk masers [115]. The follow-up imaging of these candidates has increased the number of H$_2$O maser disks with high precision VLBI maps by a factor of $\ge 4$ over the past decade. The resultant Hubble constant (see Fig. 13) from this effort is $H_0 = 73.9 \pm 3.0 \text{ km s}^{-1} \text{Mpc}^{-1}$ [110], a $\sim 4\%$ uncertain $H_0$ estimate determined based on six “clean” disk maser systems that have the required spectral qualities for reliable disk modeling. This maser-based $H_0$ measurement is well consistent with the majority of direct, late Universe measurements of Hubble constant. Its uncertainty is currently dominated by measurement errors in the maser position obtained from VLBI observation and in the determination of the maser acceleration with the single-dish monitoring. The systematic uncertainties, which could result from non-circular orbits of the masing gas, e.g., see Ref. [108] or from the impacts of non-gravitational forces such as shocks in the maser disk, e.g., see Ref. [115–117], are currently negligible in comparison with measurement errors. It is expected that the accuracy of the maser-based $H_0$ measurement can be further improved to $\sim 1\%$, the long-term goal of the observational cosmology community, by measuring distances to $\ge 50$ maser galaxies, with $\sim 7\%$ accuracy per measurement [118]. It is promising that this goal can be realized after the full array operation of the next-generation Very Large Array (ngVLA), which will bring about an order of magnitude improvement in sensitivity, permitting efficient detection of disk maser systems in a $\sim 30$ times larger volume compared with the MCP [118] as well as significant improvement in VLBI maser position measurement.

### 2.1.4. On the tip of the red giant branch method

**Coordinator:** Richard I. Anderson

**Contributors:** Adam Riess, Gagandeep S. Anand, Giulia de Somma, Ippocratis Saltas, Louise Breuval, Siyang Li, and Vladas Vansevicus

The TRGB method provides the most precise stellar alternative to classical Cepheids (Section 2.1.2) on the first and second rungs of the extragalactic distance ladder used to measure the Hubble constant [119]. Comparison between TRGB and Cepheid distances in supernova hosts shows very good agreement between both methods, Fig. 11 in Refs. [16] or [26]. Fig. 14 illustrates two paths to $H_0$ in which

<footnote n="1">¹ Here, *J* is the total angular momentum of the H$_2$O molecule, with *K*$_{-}$ and *K*$_{+}$ representing the projections of *J* on two molecular axes, e.g., see Ref. [101].</footnote>
<figure><img src="image_18.png" /><figcaption>**Fig. 13.** Left panel: Hubble diagram for the six maser galaxies considered in the MCP $H_0$ measurement [110]. Assuming a fixed velocity uncertainty of 250 km s$^{-1}$ associated with peculiar motions, the MCP constrains the Hubble constant to be $H_0 = (73.9 \pm 3.0)$ km s$^{-1}$Mpc$^{-1}$, independent of distance ladders and the CMB. Right panel: posterior distributions for $H_0$ from the five different peculiar velocity treatments considered in Ref. [110], with the fiducial treatment plotted in black.</figcaption></figure>

the TRGB calibrates either SBF (Section 2.1.5) [22,120] measured in elliptical galaxies or SNIa (Section 2.1.7), e.g., see Ref. [15,121,122] measured in any type of galaxy for tracing the Universe's isotropic expansion in the Hubble flow. In both cases, the TRGB serves as an intermediary to translate the relative apparent magnitude differences of SBF or SNIa as a function of redshift to an absolute scale anchored to geometrically measured distances. As part of the past decade's unfurling Hubble tension, the TRGB method has seen major improvements and inspired deep investigations into systematics of late-Universe $H_0$ measurements. Here, we briefly review the astrophysical basis of the TRGB as a standard candle to determine luminosity distances, present methodological considerations relevant for determining TRGB distances to better than ~5%, and consider likely future developments relevant for the $H_0$ tension in light of catalysts provided by new observational facilities and upcoming space missions. [123] provides further background and in-depth discussion.

#### 2.1.4.1. The astrophysical basis.
The TRGB as a standard candle is a robust empirical concept that is usefully supported by a solid understanding of the evolution of RGB stars. Empirically, the magnitude $m_T$ is measured as the inflection point of a RGB LF [126,127]. Measuring $m_T$ therefore requires a large number of stars to avoid stochastic effects and, in particular, cannot be done for each star separately. This characteristic is shared with the J-region AGB method [128,129] (Section 2.1.8) and distinguishes the TRGB as a statistical standard candle from pulsating stars, such as classical Cepheids, whose luminosity can be measured and calibrated individually [47,130].

The TRGB feature is astrophysically explained by the very rapid thermonuclear ignition of degenerate helium cores of low-mass first-ascent RGB stars, the so-called helium flash (HeF), see e.g., [133–137]. In particular, old RGB with masses $\lesssim 1.6 M_\odot$ develop helium cores of nearly equal mass (~0.5 $M_\odot$), resulting in a nearly equal luminosity of the HeF that can be exploited as a standard candle. Younger, higher-mass giants reach lower luminosity due to lower electron degeneracy at higher core temperatures and lower densities. This mass limit corresponds to an age-effect: giants older than ~4 Gyr have nearly constant luminosity in I-band. Differences in chemical composition (metallicity) significantly affect TRGB luminosity and temperature due to line blanketing: the bolometric magnitude behaves as $M_{T,Bol} \propto -0.19[\text{Fe/H}]$ [136]. At the same time, higher metallicity results in cooler effective temperature. Hence, more metal-poor giants are brighter and bluer. Thankfully, the dependence on metallicity is partially mitigated in Cousins I-band where the bolometric correction ($M_I = M_{Bol} - BC_I$) nearly compensates the change in luminosity for low-metallicity ([Fe/H] $\in$ [-0.7, -2.0]) giants. Color-based metallicity calibrations can further mitigate this issue, e.g., see Ref. [138,139]. Hence, the TRGB

is a theoretically well-supported standard candle when low-metallicity ([Fe/H] < -0.7) old (> 4 Gyr) RGB stars are observed in I-band. However, observations at infrared wavelengths require caution and additional study with respect to the aforementioned effects, in particular if the observed RGB populations exhibit diversity in age and/or chemical composition, see Fig. 15.

Theoretical predictions of the TRGB luminosity rely on stellar evolution models that require various assumptions and simplified treatment of stellar physics, including the modeling of opacities, diffusion, convection (mixing-length theory) and overshooting, mass-loss, electron screening, neutrino losses [140], nuclear reaction rates, among others [137,141]. In particular, the dominant uncertainties related to radiative opacity introduce systematics of up to 1.6%. Simultaneously varying radiative and conductive opacities, nuclear reaction rates (e.g., triple-alpha reaction), and neutrino losses affect the bolometric Tip luminosity by $\delta L/L \sim 10^{-3}$, depending on mass and metallicity [142]. Comparisons with observations are subject to larger uncertainties, notably related to the translation of bolometric luminosity to magnitude, which require stellar atmosphere models that are usually considered separately from the evolutionary models [137].

#### 2.1.4.2. Methodological considerations and HST-based results.

The absolute calibration of the TRGB method requires measuring $m_T$ in RGB populations of known distance to obtain $M_T = m_T - \mu_0$. The best available, geometrically measured, distances (in ascending order) are: trigonometric parallaxes from the ESA Gaia mission [143], the distances to the LMC and SMC measured by detached eclipsing binary stars [29,30], and the megamaser distance to NGC 4258 (M106) [31]. Importantly, the distance moduli $\mu_{0,i}$ of the LMC, SMC, and NGC 4258 are frequently used in distance ladders calibrated using stellar standard candles and hence their systematics are often shared among different $H_0$ measurements, e.g., see Ref. [34,35,144,145]. Comparing distances to other galaxies measured using an absolute calibration based on the same anchor, such as NGC 4258, thus yields the strongest test of distance systematics [e.g.,26,146]. As implied in Fig. 14, uncertainties in $\mu_0$ propagate directly into $H_0$ measurements, as they set the absolute scale of the secondary tracers (SNIa, SBF) via the primary standard candle (here: TRGB). Basing an absolute TRGB calibration on the broadest possible set of “anchors” reduces possible bias and allows one to determine appropriate standardization procedures, e.g., to account for metallicity differences [121,138]. At present, the most accurate TRGB calibrations to date are obtained in the Magellanic Clouds [126,147] and NGC 4258 [16] using passbands similar to I-band, such as HST’s ACS/F814 W or JWST’s NIRCAM/F090 W. TRGB calibration based on Gaia parallaxes [148] has been achieved based on EDR3 parallaxes, which require correction for complex systematics, e.g., see Ref. [77,
## Surface Brightness Fluctuations

Type-Ia supernovae (SNela)

<figure><img src="image_19.png" /><figcaption>**Fig. 14.** Illustrations of extragalactic distance ladders built on the TRGB method. The scales considered increase upwards. **Bottom right:** The TRGB feature is calibrated based on color-magnitude diagrams of stars whose distances ($\mu_0$) are known from geometric measurements, such as Gaia parallaxes, the Magellanic Clouds, or NGC4258. The choice and systematics of (the combination of) geometric anchors directly impact the Hubble constant by setting $\mu_0$. **Center left:** The apparent magnitude of the TRGB feature is determined in other galaxies. Differences in the absolute magnitude between TRGB populations, either compared to the calibrating set or as field-to-field variations, must be mitigated by standardization. **Upper part, right:** The measured TRGB distances calibrate the fiducial absolute magnitude of SNIa in any type of galaxy. Cross-checks between multiple stellar standard candles can primarily be obtained in galaxies hosting young, intermediate-age, and old stellar populations at the same time. Note that TRGB fields should be placed on galaxy halos populated by old, metal-poor stars, whereas classical Cepheid fields target young stellar populations in supernova host galaxies. **Upper part, left:** Alternatively, the TRGB is used to calibrate the absolute scale of the SBF method for distance determination to elliptical galaxies that trace the Hubble flow. SBF exploits the fact that the variance of SBF decreases as $d^2$.</figcaption></figure>

Figure credit: R.I. Anderson based on images as labeled from: J+21 [124]; F+11 B. J. Fulton, Las Cumbres Observatory; R+22 [34]; W+23 [125], A+24 [126].

78,83,149–152]. Furthermore, a parallax-based TRGB calibration has been obtained for ωCentauri [153]. The period–color relation of stars near the TRGB may provide a useful avenue to deal with (differential) reddening when calibrating the TRGB based on field RGs [147].

A critical element of determining accurate distances using standard candles is to ensure consistency between the absolute calibration and the standardized application in the target environments. However, there are measurable differences even between the I-band absolute magnitudes in the LMC and the SMC, and they reflect noticeably in the variability periods of the small-amplitude RGB stars that make up the RGB population at the Tip [126]. These ubiquitous small amplitude pulsations near the RGB Tip allow us to probe the intrinsic diversity (e.g., in age and metallicity) of RGB populations that is not typically known a priori and thus constitute a difficult-to-control astrophysical systematic at the level of a couple of percent [147]. Additionally, a common feature of RGB populations in other galaxies is contamination by other stars, notably at higher luminosity (typically AGB stars).

Unfortunately, the specifics and degree of such contamination differ from environment to environment (cf. CATs below) and are not reliably known a priori. The exception to this rule is globular clusters (GC), which rarely contain stars brighter than the TRGB, e.g., see Ref. [154], so that measuring $m_T$ conceptually corresponds to searching for the brightest cluster star without knowing how much it differs from the HeF luminosity. As a result, GC-based values of $m_T$ represent lower limits to the desired measurement of $m_T$ and are furthermore also subject to variations in metallicity and age, depending on the photometric band. Thus, GC-based $m_T$ values differ conceptually from the values of $m_T$ measured in RGB populations of mixed age and metallicity. Further study is needed to assess to what degree such effects impact $M_T$, and hence, $H_0$.

Extinction corrections are routinely applied for TRGB calibration in the Magellanic Clouds [155]. However, dust corrections to TRGB measurements in other galaxies typically, e.g., see Ref. [144] rely on
<figure><img src="image_20.png" /><figcaption>**Fig. 15.** Dependence of $M_T$ on chemical composition and age based on PARSEC v.1.2S isochrones [131,132]. *Top:* 8 Gyr isochrones for increasing metallicity (from left to right: [M/H] = -2.2, -1.5, -0.7, -0.4, 0.0) in Johnson V (dark blue), Cousins I (cyan), and JWST/NIRCAM F150 W (red) passbands. *Bottom:* [M/H] = -1.5 isochrones for increasing ages (left to right) of 2 Gyr (blue), 4 Gyr (cyan), 8 Gyr (green), and 14 Gyr (red) in the Johnson V, Cousins I, JWST/NIRCAM F090 W and F150 W passbands, successively offset in color for visibility. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

all-sky dust extinction maps [156,157] that account only for MW foreground dust and whose accuracy is limited in the vicinity of resolved galaxies. Recently, it has been pointed out [158] that small, albeit non-zero, extinction in galaxy halos leads to underestimated $H_0$ values and that dust extinction estimates based on background quasars [159] provide a possibility for correcting this one-sided systematic. Statistical corrections for local extinction have since been applied [16,122]. However, further study of extinction effects and their variation across circumgalactic media would be useful to provide improved corrections.

The TRGB magnitude ($m_T$) is measured either via an edge detection algorithm [119,127], such as a Sobel filter, or via a maximum likelihood fit of the LF [79,160,161]. When Sobel filters are used for edge detection, LFs are typically smoothed to some degree to reduce noise [127], and different weighting schemes have been considered in the literature [125,162]. Recently, it has been pointed out that both the LF smoothing and the Sobel response weighting introduce biases that depend on the shape of the observed LFs [126]. In particular, Sobel response weighting was shown to introduce the tip-contrast relation determined independently using observations [125]. Both issues can easily bias the measured value of $m_T$ by 0.06 mag (~ 3% in distance), and it is therefore crucial to apply the same TRGB measurement method in all target RGB populations to avoid bias. Similarly, observer choices, such as color cuts (notably near the Tip), can affect the measured $m_T$ and should be decided based on objective criteria [122].

### Field-to-field variations of $m_T$

have been recently reported within the same galaxies. This led to the development of an unsupervised TRGB detection algorithm, called CATS (Comparative Analysis of TRGBs), which sought to reduce the impact of subjective observer choices on the measurement of $m_T$ [122,125]. In the process, a tip-contrast relation was determined, which has since been shown to result from the use of weighted Sobel response curves [126]. Bias of TRGB distances can thus be avoided if either (a) unweighted Sobel filters are used to measure $m_T$ [126], or (b) if an appropriate tip-contrast relation is used to standardize $m_T$ a posteriori when weighted Sobel response curves are used [122]. Given the simpler algorithm and the non-uniqueness of the tip contrast, which depends on the measured $m_T$, it appears prudent to avoid Sobel filter weighting. First results from an I-band (F090 W) TRGB calibration using the JWST also support this approach [16].

In observations based on the Hubble Space Telescope or JWST, the placement of TRGB fields in target galaxies is crucial to ensure that the targeted RGB population is similar or standardizable to the RGB population that provides the absolute calibration [163], for an example, e.g., see Ref. [146]. Thanks to very large fields-of-view, observations with the ESA Euclid mission [164] or the future Nancy Grace Roman telescope<fnref n="2" /> will allow us to integrate field-to-field variation analysis into TRGB algorithms, rendering them much more robust. However, the existence of field-to-field variations within a single galaxy highlights the need for TRGB standardization to account for differences between RGB populations in different galaxies.

### The choice of photometric system

The TRGB has been most commonly measured in the I-band where it is nearly flat and relatively insensitive to metallicity and age effects [119,145], cf. Fig. 15. There has been growing interest in NIR TRGB measurements due to the capabilities of the JWST and the intrinsically higher (1–2 mag) luminosity in the NIR [165,166], and several I-band and NIR observing programs are currently being analyzed with JWST (e.g., GO-1685, 1995, 2875, 3055). However, the (color) slope of the TRGB at NIR wavelengths requires additional consideration. Both empirical (see Refs. [165–172]) and theoretical approaches [137,173] have been considered for the time being, and further study is required to determine competitive and accurate TRGB distances at infrared wavelengths. Furthermore, K-corrections are expected to exceed 1% in distance for single-band JWST TRGB observations at distances above ~70 Mpc for the F150 W passband and shorter wavelengths [158].

#### 2.1.4.3. Implications for $H_0$ and the Hubble constant tension.
Several studies have reported $H_0$ measurements involving the TRGB as a standard candle, with rather significant differences. To understand these differences, it is crucial to consider the differing assumptions and methods underlying each analysis. In particular, Refs. [121,144] applied different absolute TRGB calibrations and TRGB measurement methodologies to the same set of HST I-band TRGB observations in conjunction with the CSP SNIa dataset in the Hubble flow and measured $H_0 = 69.8 \pm 0.6 \pm 1.6 \text{ km s}^{-1} \text{Mpc}^{-1}$ and $71.5 \pm 1.8 \text{ km s}^{-1} \text{Mpc}^{-1}$, respectively. In an update to Ref. [144], Ref. [174](v3) employed JWST I-band (F090 W) TRGB measurements together with CSP SNIa and obtained $H_0 = 70.4 \pm 1.2 \pm 1.3 \pm 0.7 \text{ km s}^{-1} \text{Mpc}^{-1}$. Ref. [122] corrected the tip-contrast TRGB systematic and replaced the CSP SNIa with Pantheon+ to ensure a consistent calibration between SNIa in host galaxies and the Hubble flow and found $72.94 \pm 1.98 \text{ km s}^{-1} \text{Mpc}^{-1}$. Replacing HST entirely with JWST I-band TRGB observations of 17 unique SNIa yields $72.1 \pm 2.2 \pm 1.2 \text{ km s}^{-1} \text{Mpc}^{-1}$ [26]. Recently, Ref. [120] replaced SNIa entirely with SBF calibrated by JWST I-band TRGB observations [16] and found $73.8 \pm 0.7 \pm 2.3 \text{ km s}^{-1} \text{Mpc}^{-1}$, confirming the conclusion from Ref. [122] that at least half the difference between the $H_0$ values reported by Ref. [144] and Ref. [82] can be attributed to SNIa alone.

<footnote n="2">² https://roman.gsfc.nasa.gov</footnote>
The prospects for further improvements of the TRGB method and, in turn, to increase the accuracy in $H_0$ are very promising. New wide-field space-based telescopes (Euclid, Roman) will allow us to comprehensively measure $m_T$ in a very large number of galaxies, while JWST will allow to push the limits of TRGB distances. Potentially, *Roman* could allow to directly calibrate $m_T$ in the LMC and SMC, circumventing differences among photometric systems. However, the target magnitudes are close to saturation (~6s at AB 15 mag in F087$^3$). An SBF calibration based on the TRGB method will yield an accurate $H_0$ measurement fully independent of the Cepheid-SNIa distance ladder [22,120], if different geometric distances are used to calibrate the two types of standard candles. Future improvements in the *Gaia* astrometric solution will play an important role in this endeavor, as will synthetic *Gaia* photometry based on $B_p$ and $R_p$ spectra [175]. Pushing the TRGB to the limit using JWST may motivate infrared observations owing to the combination of RGB stars being brighter in the near-IR and JWST providing optimal sensitivity there, although it has been already shown that I-band JWST TRGB measurements are feasible out to at least 50 Mpc [22,176].

In order to ensure that stellar standard candles provide the clearest and most accurate picture of cosmic tensions, notably the Hubble constant tension, it is crucial to pursue the most direct and simple assessments of systematics. To this end, direct and detailed comparisons of distances measured using the TRGB method, classical Cepheids, and the JAGB method are preferred, cf. [26]. In contrast, trying to understand TRGB or Cepheid-based systematics from published values of $H_0$ is complicated by several other possible differences between distance ladder set ups, e.g., with respect to the treatment of peculiar motions and supernova standardization [122]. Cross-checks based on nearby galaxies, e.g., M31 [85], M33 [46,177], and other nearby galaxies (e.g., HST GO-17520), will be particularly insightful to this end as they provide the greatest possible precision for each of the standard candles and an optimal ability to investigate causes of systematic differences, e.g., using spectroscopy on future 30m-class telescopes, such as the ELT.

### 2.1.5. The surface brightness fluctuations method

**Coordinator:** Michele Cantiello

**Contributors:** Enzo Brocato, Gabriella Raimondo, John Blakeslee, Joseph Jensen, and Rebecca Habas

#### 2.1.5.1. Introduction & state of the art.
The SBF method is a powerful technique used to measure distances of elliptical galaxies out to ~150 Mpc, and may reach distances as far as ~300 Mpc with telescopes like the JWST [22]. Introduced in the late 1980s, the SBF method relies on the fact that the surface brightness of a galaxy exhibits small-scale variations due to the statistical distribution of stars [178–182]. When imaging a galaxy, the number of stars that fall within each detector pixel varies, causing pixel-to-pixel variations in the surface brightness. The amplitude of these fluctuations is inversely proportional to the square of the distance to the galaxy [183], allowing one to estimate distances from imaging alone.

In principle, the SBF method is straightforward. The only difficulty lies in isolating the signal fluctuations of the underlying Population II stars from contaminants such as globular clusters, star forming regions, and background galaxies. Once this is done, the absolute SBF magnitude can be calibrated in a given passband, $\bar{M}_{\xi}$, using galaxies at well-known distances [184] or through stellar populations numerical modeling [185]. The distance modulus $\mu_0 \equiv (m-M) = 5 \log 10(D) + 25$ can then be inferred as usual: $\mu_0 = \bar{m}_{\xi} - \bar{M}_{\xi}$ where $\bar{m}_{\xi}$ is the $\xi$-band observed fluctuation magnitude and D is the distance in Mpc.

In practical terms, the SBF distance measurement involves several steps, summarized briefly here and illustrated in Fig. 16: (i) determining and subtracting the sky background; (ii) modeling and subtracting the

galaxy (upper middle panel in Fig. 16); (iii) masking all potential sources of contamination to the fluctuation signal (e.g., globular clusters within the galaxy, background galaxies, dust patches, etc.; upper right panel); (iv) modeling the LF of sources in the frame to estimate the spurious fluctuation contribution from unexcised sources (lower left panel); and (v) power spectrum analysis of the residual masked frame, to measure the amplitude of the SBF signal (lower middle and right panels). Additional details on this procedure are available in the references cited in this review.

In recent years, the application of the SBF method has seen significant advancements, particularly with the contributions from the high-resolution space-based data from the HST. Recently, Refs. [21, 124] used HST/WFC3 data to measure highly accurate distances for a sample of ~60 elliptical galaxies, taking particular care to analyze the $\bar{M}_{\text{F110W}}$ calibration, and succeeded in obtaining distances with median statistical uncertainties $\le 4\%$.

From the ground, the Next Generation Virgo Cluster Survey (NGVS), a project based on 104 degrees² deep optical imaging from the CFHT [187], has provided a dataset ideal for SBF measurements in the Virgo Cluster. The NGVS has enabled detailed studies of galaxy distances in this cornerstone galaxy cluster [186], and represents a crucial survey, serving as a precursor for future large ground- and space-based surveys.

Most of the focus for the use of SBF has been on bright galaxies, where the method provides the lowest uncertainty, reaching levels $\le 5\%$ on single targets in well-designed observations. The advent of large area deep surveys has, however, led to the discovery of an increasingly large number of dwarf galaxies which play a fundamental role in Cosmology [188]. Several authors [189–191] have applied the SBF technique to smaller and fainter systems. These studies have demonstrated that $\bar{m}$ can be used to estimate distances to dwarf galaxies, although with relatively large errors compared to massive galaxies, providing a typical uncertainty of $\delta D/D \sim 15\%$. Despite the relatively larger error, these elusive galaxies are often challenging targets for distance measurement due to their low surface brightness and complex stellar populations. This has opened new avenues for using SBF to probe the Local Universe and improve the cosmic distance ladder.

#### 2.1.5.2. The Hubble constant tension.
In addition to providing distance estimates for individual galaxies, SBF can also be used to probe tensions in the measurement of the Hubble constant that have arisen between the CMB observations [192] and the value obtained from the late-Universe [34].

The SBF method, with its ability to provide distance measurements to galaxies beyond the 50 Mpc range, offers a unique perspective on this tension, independent of the classical Cepheid/SNIa route. This topic has been explored by Ref. [21] for a sample of bright galaxies between 20–100 Mpc, with $\bar{M}_{\text{F110W}}$ calibrated using both Cepheids and TRGB distances. For both calibrations, the authors derived a value of $H_0$ that is fully consistent with the direct distance estimates in the local Universe. The work by Ref. [21] was based on a small sample, and this result should be corroborated with larger studies. That study, as well as some of the newly programmed ones [22], used SBF in near-IR passbands where the amplitude of fluctuations is “enhanced” because the brightest stars contributing to the signal are brighter compared to the optical bands. This enhancement is key for the recent results on $H_0$ and will be crucial for future observations, as it allows for reliable measurements for galaxies at larger distances.

A complementary approach to the $H_0$ tension uses SBF indirectly, as a calibration for SNIa distances. Rather than the Cepheid-SNIa route to Cosmology, one can use a Cepheid-SBF-SNIa route. This method has larger systematic and statistical uncertainties from the extra rung in the distance ladder, but offers the advantage of having a larger calibration sample of SNIa from the local Universe because of the larger overlap in galaxies that host SNIa and have SBF measurements. This approach was adopted by Ref. [193], who used a heterogeneous

<footnote n="3">³ https://roman.gsfc.nasa.gov/science/apttables2021/table-timetosaturation.html</footnote>
<figure><img src="image_21.png" /><figcaption>**Fig. 16.** SBF analysis for the galaxy VCC1146. **Top row:** i-band image, residual image, and masked residual image (left to right). Black annuli in the second and third panels mark the inner and outer radii used for SBF measurements. **Bottom left:** Fitted LF for external sources. Green circles are observational data with a faint-end downturn from incompleteness. The solid green line shows the best fit, corrected for incompleteness, with blue dotted and red dashed curves representing the background galaxy and GC LFs. **Bottom middle:** Azimuthal average of the residual power spectrum (gray circles) and the fit (solid black line). **Bottom right:** Fitted $P_0$ values vs. $k_{\text{start}}$, with $k_{\text{end}} = 450$. Vertical lines at $k_{\text{start}} = 80$ and 160 show the range over which the fit is stable. The median $P_0$ in this range (dotted horizontal line) is adopted.

*Source:* Figure adapted from Ref. [186].</figcaption></figure>

collection of SNIa and SBF measurements from existing literature, obtaining $H_0 = 71.1 \pm 2.4(\text{stat}) \pm 3.4(\text{sys}) \text{km s}^{-1}\text{Mpc}^{-1}$. More recently, Ref. [194] used a very homogeneous sample of SBF measurements from Ref. [124], and found that the Hubble–Lemaître parameter derived from the revised SALT2 [32] parameter fit and calibrated with near-IR SBF distances is $H_0 = 73.3 \pm 1.0(\text{stat}) \pm 2.7(\text{sys}) \text{km s}^{-1}\text{Mpc}^{-1}$. Note that the SNIa-independent estimate from Ref. [21], based on Cepheid zero-point calibration, yielded $H_0 = 73.3 \pm 0.7(\text{stat}) \pm 2.4(\text{sys}) \text{km s}^{-1}\text{Mpc}^{-1}$, whereas the most recent recalibration, which relies on TRGB distances from JWST observations of Virgo and Fornax galaxies by Ref. [120], provided $H_0 = 73.8 \pm 0.7(\text{stat}) \pm 2.3(\text{sys}) \text{km s}^{-1}\text{Mpc}^{-1}$.

### 2.1.5.3. Future prospects and projects.
The SBF method is expected to become even more relevant in the future, taking advantage of the fact that most of the forthcoming facilities will operate in the near-IR regime, where the SBF signal is much stronger. The JWST, with its superior resolution and sensitivity, is expected to enable detailed SBF studies of distant galaxies, potentially reaching ~300 Mpc [22]. The large sky surveys from the Vera Rubin Observatory's LSST and from the Euclid mission, will further enhance the capabilities of SBF by providing an enormous dataset for SBF, largely dominated by the dwarfs. This will allow astronomers to apply the SBF method to a much larger and more diverse sample of galaxies, improving the statistical robustness of the 3D mapping of the Universe within 40–70 Mpc.

On longer time scales, the Nancy Grace Roman Space Telescope (to be launched in the mid-late 2020s), will also play a crucial role in the advancement of SBF, due to its combination of wide-field, high-resolution, and infrared wavelength coverage. Roman will enable high-precision SBF measurements over large areas of the sky, potentially reaching the depth of the JWST, but over a wider area. The future 30–40 m class ground based telescopes, with their adaptive optics (AO), may also be more useful for SBF than the JWST. Correcting for atmospheric distortions using the AO systems could provide better

FWHM resolutions than the JWST in certain passbands (e.g., K-band) which, combined with the huge telescope collecting area, would allow SBF to be applied to more and more distant systems. Although time-consuming, because of the specific requests of AO observations of extended objects and the needs for AO activation loops, observing a handful of well-chosen targets with one of these massive telescopes could allow one to discriminate between different cosmological models, independently from SNIa.

Beyond improved telescopes, SBF observations also benefit from improved numerical stellar population studies. Accurate models of stellar populations are essential for interpreting SBF measurements, reducing systematic uncertainties and deriving reliable k-corrections which cannot be ignored beyond ~150 Mpc. Additionally, the application of SBF to blue, low-mass galaxies requires the calibration of $\bar{M}$ across a broader range of stellar populations and environments. Stellar population modeling will offer a reference to test and prove the reliability of the SBF method over a wide interval of population properties.

Theoretical efforts over the past 20 years have focused on incorporating stars in peculiar evolutionary stages, such as thermally pulsing AGB stars and hot HB stars in old-intermediate populations (e.g., see Ref. [185,195]). These efforts have also included the impact of α-elements and He-abundance enhancements. Advancements in this area will benefit from the insights gained by applying theories of late stellar evolution — dominated by poorly understood physics such as mass loss processes — to the observations in mid-infrared instruments like MIRI on the JWST. This will result in more refined stellar population models for distance estimates and a better understanding of these crucial stages of stellar evolution.

Future SBF measurements can also benefit from the adoption of ML techniques. ML methods require training on large samples that are not yet available, but when they are, ML algorithms can optimize the measurement of SBF, which is currently slow and heavily reliant on human