“…In this paper, we examine a cosmological vector field non-minimally coupled to gravity through the Gauss-Bonnet invariant. Although couplings between the Gauss-Bonnet invariant and scalar fields have been explored in the context of inflationary cosmology (Tsujikawa, Sami, & Maartens, 2004;Carter & Neupane, 2006.;Satoh & Soda, 2008;Guo & Schwarz, 2009;Sadeghi, Setare, & Banijamali, 2009;Guo & Schwarz, 2010;Satoh, 2010;Jiang, Hu, & Guo, 2013;Nozari & Rashidi, 2013;Koh, Lee, Lee, & Tumurtushaa, 2014;Okada & Okada, 2016;Kanti, Gannouji, & Dadhich, 2015;van de Bruck, Dimopoulos, & Longden, 2016;Koh, Lee, & Tumurtushaa, 2017;Mathew & Shankaranarayanan, 2016;van de Bruck, Dimopoulos, Longden, & Owen, 2017;Fomin & Chervon, 2017;Yi, Gong, & Sabir, 2018;Granda & Jimenez, 2019b, 2019aJimenez, Granda, & Elizalde, 2019;Kleidis & Oikonomou, 2019;Fomin, 2020;Pozdeeva, 2020;Rashidi & Nozari, 2020), the coupling with a massive vector field has not been sufficiently explored in the literature (Oliveros, 2017). Non-minimal couplings of the electromagnetic field to gravity, in particular to the Gauss-Bonnet invariant, have been considered as a mechanism to generate large-scale magnetic fields during inflation (Sadeghi et al, 2009;Bamba & Odintsov, 2008).…”