Is there indirect selection on female extra-pair reproduction through cross-sex genetic correlations with male reproductive fitness?

One key hypothesis explaining the evolution and persistence of polyandry, and resulting female extra-pair reproduction in socially monogamous systems, is that female propensity for extra-pair reproduction is positively genetically correlated with male reproductive fitness and consequently experiences positive cross-sex indirect selection. However, key genetic correlations have rarely been estimated, especially in free-living populations experiencing natural (co)variation in reproductive strategies and fitness. We used long-term life-history and pedigree data from song sparrows (Melospiza melodia) to estimate the cross-sex genetic correlation between female propensity for extra-pair reproduction and adult male lifetime reproductive success, and thereby test a key hypothesis regarding mating system evolution. There was substantial additive genetic variance in both traits, providing substantial potential for indirect selection on female reproductive strategy. However, the cross-sex genetic correlation was estimated to be close to zero. Such small correlations might arise because male reproductive success achieved through extra-pair paternity was strongly positively genetically correlated with success achieved through within-pair paternity, implying that the same successful males commonly sire offspring produced by polyandrous and monogamous females. Cross-sex indirect selection may consequently have limited capacity to drive evolution of female extra-pair reproduction, or hence underlying polyandry, in systems where multiple routes to paternity success exist.