Comment on "Late-time tails of a self-gravitating scalar field revisited" by Bizon et al: The leading order asymptotics

In arXiv:0812.4333v2 Bizon et al discuss the power-law tail in the long-time evolution of a spherically symmetric self-gravitating massless scalar field in odd spatial dimensions. They derive explicit expressions for the leading order asymptotics for solutions with small initial data by using formal series expansions. Unfortunately, this approach lacks insight into the very origin of the leading asymptotics and misses an interesting observation that the final decay rate results from very special cancellations in the nonlinear terms. Here, we show that one can avoid tedious manipulations of series and calculate the leading asymptotics more directly by recognizing the special structure and cancellations already on the level of the wave equation