deletions | additions       diff --git a/sectionDiscussion_an.tex b/sectionDiscussion_an.tex  index 8825414..24e49b3 100644  --- a/sectionDiscussion_an.tex  +++ b/sectionDiscussion_an.tex 
...
If AM is conserved during the supernova, stochastic wave torques entail that most NS are born with $P \lesssim 1 {\rm s}$. Although there is significant uncertainty, our best estimates yields entail typical NS spin periods at birth of $P \sim {\rm few} \times 100 \, {\rm ms}$. This is comparable to spin periods of some young, slowly rotating NSs (\citealt{lai:96,gotthelf:13}), and for the broad inferred birth spin period distribution of $P \lesssim 500 \, {\rm ms}$ for ordinary pulsars (\citealt{faucher:06,popov:10,gullon:14}). Therefore, stochastic wave spin-up could be the dominant mechanism in determining the rotation periods of pre-collapse SN cores and newborn NSs.  There is ample evidence that {\it some} CC events occur with rapidly rotating cores. In particular, long GRBs almost certainly require a rapidly rotating central engine \citep{Woosley_1993,Yoon_2006,Woosley_2006,Metzger_2011}, and the picture advanced above must break down in certain (although somewhat rare) circumstances. It is not immediately clear what factors contribute to the high spin rate in GRB progenitors, as our analysis was restricted to ``typical" NS progenitors with $M \sim 12 $10 M_\odot \lesssim M lessim 20  M_\odot$, which explode to produce type-IIp supernovae during a red supergiant phase \citep[See e.g.][]{Smartt_2009}. We speculate that GRB progenitors have {\it never} undergone a red supergiant phase, as torques via magnetic fields and/or waves are likely to spin-down the helium core by coupling it with the huge AM reservoir contained in the slowly rotating convective envelope. Alternatively, it may be possible that stars with very massive He cores, which exhibit more vigorous pre-SN burning phases, can generate stochastic wave spin-up strong enough to produce a GRB. The population of massive stars approaching death is complex, and factors such as initial mass, rotation, metallicity, binarity, magnetic fields, overshoot, mixing, winds, etc., will all contribute to the anatomy of aging massive stars. We have argued that AM transport via convectively driven waves is likely to be an important factor in most massive stars. But it is not immediately obvious how this picture will change in different scenarios, e.g., electron capture supernovae, very massive $(M_i \gtrsim 40 M_\odot)$ stars, interacting binaries, etc. We hope to explore these issues in subsequent works.