The development and spread of agriculture changed fundamental characteristics of human societies1-3. However, the degree to which environmental and social conditions enabled the origins of agriculture remains contested4-6. We test three hypothesized links between the environment, population density and the origins of plant and animal domestication, a prerequisite for agriculture: (1) domestication arose as environmental conditions improved and population densities increased7 (surplus hypothesis); (2) populations needed domestication to overcome deteriorating environmental conditions (necessity hypothesis)8,9; (3) factors promoting domestication were distinct in each location10 (regional uniqueness hypothesis). We overcome previous data limitations with a statistical model, in which environmental, geographic and cultural variables capture 77% of the variation in population density among 220 foraging societies worldwide. We use this model to hindcast potential population densities across the globe from 21,000 to 4,000 years before present. Despite the timing of domestication varying by thousands of years, we show that improving environmental conditions favoured higher local population densities during periods when domestication arose in every known agricultural origin centre. Our results uncover a common, global factor that facilitated one of humanity's most significant innovations and demonstrate that modelling ancestral demographic changes can illuminate major events deep in human history.