Why might the number of live births have begun to decline in the late 1950's

The baby booming age was over. Every one was back from the war and had already had kids.

Since the beginning of the age of the Enlightenment and over the course of modernization, the mortality of children below 5 years of age has declined rapidly. Child mortality in rich countries today is much lower than 1%. This is a very recent development and was only reached after a hundredfold decline in child mortality in these countries. In early-modern times, child mortality was very high; in 18th century Sweden every third child died, and in 19th century Germany every second child died. With declining poverty and increasing knowledge and service in the health sector, child mortality around the world is declining very rapidly: Global child mortality fell from 18.2% in 1960 to 4.3% in 2015; while 4.3% is still too high, this is a substantial achievement.