\section{Discussion and Conclusions}


We have shown that convectively generated internal gravity waves (IGW) in massive stars are capable of redistributing angular momentum (AM) on short time scales. In particular, inwardly propagating IGW launched from convective shells may be able to slow down the core to much slower spin rates than would be obtained in the absence of other AM transport mechanisms. For our $12 M_\odot$ model, IGW generated at the base of the convective zone during the core He burning red supergiant phase may slow the core to minimum spin periods of $P \sim 2 \, {\rm days}$. IGW launched during C shell burning may also be able to substantially slow the spin of the core. These convective phases may pre-collapse iron cores to rotation periods $P_{\rm min,Fe} \gtrsim 50\,{\rm s}$, corresponding to initial NS rotation periods of $P_{\rm min,NS} \gtrsim 2.5 \, {\rm ms}$. 


The rotation periods listed above are minimum periods for our stellar model. Calculations of rotation rates including magnetic torques \citep{Heger_2005,wheeler:14} typically yield rotation periods several times larger. Magnetic torques may therefore be the dominant AM transport mechanism responsible for extracting AM from massive stellar cores, although it is possible that both mechanisms play a significant role. If IGW are able to spin down cores to slower rotation rates, as we have speculated, then they could be the dominant AM redistribution mechanism during a massive star's life. It may seem surprising that AM transport via IGW can act on the short stellar evolution timescales of massive stars. However, the huge convective luminosities inside evolved massive stars ensure large fluxes of IGWs (QS12, SQ14) that can transport energy and EM on short timescales. We therefore encourage efforts to incorporate the effects of IGW in stellar evolution codes focusing on the final stages of massive star evolution. 


Stochastic influxes of IGW also lead to minimum core rotation rates, which may be realized given very efficient prior core spin-down via IGW/magnetic torques. Such a spin-down is not unreasonable, especially given that the cores of low mass red giant stars rotate slower than can be accounted for using hydrodynamic mechanisms or magnetic torques via the Tayler-Spruit dynamo \citep{cantiello:14}. It is thus quite plausible that massive star cores are efficiently spun down via waves/magnetic torques, after which they are stochasticly spun up via waves launched during O/Si burning. If this mechanism determines the core spin rate before death, it entails a Maxwellian distribution in spin frequency, with typical spin periods of $300 \, {\rm s} \lesssim P \lesssim 10^4 \, {\rm s}$. We thus find it extremely unlikely that magnetic torques can enforce very large pre-collapse spin periods as claimed by \cite{spruit:98}. Additionally, we speculate that the stochastic spin-up process is relatively insensitive to binary interactions or winds that have stripped the stellar envelope. As long as these processes do not strongly modify the core structure and late burning phases, stochastic IGW spin-up of the core should be able to occur. We also express a word of caution, as Si burning is notoriously difficult for stellar evolution codes to handle, and the properties of Si burning produced by our MESA evolutions have large associated uncertainties. The rough energy and AM fluxes in convectively excited waves are, however, reasonable at the order of magnitude level.


If AM is conserved during the supernova, stochastic IGW spin-up entails NS birth periods of $20 \, {\rm ms} \lesssim P \lesssim 400 \, {\rm ms}$, albeit with significant uncertainty. These estimates are comparable to spin periods of typical 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. In this scenario, there is little or no correlation between the spin of the progenitor and the spin of the NS it spawns. Although torques during the supernova may modify the spin rate of the NS, they would have to be very finely tuned to erase the stochastic spin-up occurring during shell burning. Moreover, any sort of purely frictional spin-down processes would likely slow the NS to rotation periods larger than typically inferred for younng NSs. 


Given the discussion above, we advance a heuristic picture for the rotational evolution of the cores of massive stars. After the main sequence, the core contracts and spins up, but is quickly spun down via coupling with the slowly rotating convective envelope. The coupling is likely mediated by strong magnetic torques (likely via a fossil field or dynamo-generated field from a prior core-burning phase) as seems to be required in low-mass stars. During shell O/Si burning, however, the core is stochastically spun back up by the huge influx of convectively excited IGW. We note that this scenario is very similar to the stochastic spin-up scenario proposed by \cite{spruit:98}, except that the AM depostion occurs before core-collapse, and the source of the AM (convectively excited IGW) is somewhat better understood. An attractive feature of the stochastic spin-up scenario is that it is relatively insensitive to the evolutionary history of the star, which is riddled with complications such as birth spin period distribution, mass loss, binarity, AM redistribution, etc. In contrast, stochastic spin-up is sensitive only to the basic properties of O/Si shell burning, and may offer a much simpler avenue for predicting the spin rates of compact objects.

% We note that the AM deposited by IGW also entails a characteristic momentum deposition of $P_{\rm ex} \sim I_c \Omega_{\rm ex}/r_c$, which corresponds to a core velocity of $v_{\rm ex} \sim P_{\rm ex}/M_c$. Our calculations entail typical ``kick" velocities of $ 10^2 \, {\rm km} \, {\rm s}^{-1} \lesssim v_{\rm ex} \lesssim 7 \times 10^2 \, {\rm km} {\rm s}^{-1} $, very similar to typical NS kick velocities ***REF***. It may therefore be possible that stochastic momentum/AM deposition accounting for NS kicks/spins occurs in a similar fashion outlined in \cite{spruit:98}, but with the momentum deposition occuring {\it before} core collapse via IGW excited by vigorous shell burning. 


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{1993ApJ...405..273W,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 $10 M_\odot \lesssim M \lesssim 20 M_\odot$, which explode to produce type-IIp supernovae during a red supergiant phase \citep[See e.g.][]{Smartt_2009}. We speculate that GRB progenitors (if occurring in effectively single star systems) 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. A third possibility is that spin-up via mass transfer/tidal torques in binary systems is required for GRB production \cite{Cantiello_2007}.



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 IGW 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. 
