\section{Introduction}
\label{intro} 

Rotation is a key player in the drama that unfolds upon the death of a massive star. The angular momentum (AM) contained in the iron core and overlying layers determines the rotation rate at core collapse (CC), which could have a strong impact on the dynamics of CC and the subsequent supernova \citep[See e.g][]{MacFadyen_1999,Woosley_2002,Woosley_2006,Yoon_2006}. Rotation may help determine the nature of the compact remnant, which could range from a slowly rotating neutron star (NS) to a millisecond magnetar or rapidly rotating black hole \citep[See e.g.][]{heger:00,Heger_2005}. The former may evolve into an ordinary pulsar, while the latter two outcomes offer exciting prospects for the production of long gamma-ray bursts (GRB) and superluminous supernova. In each of these phenomena, rotating central engines are suspected to be the primary source of power \citep{1993ApJ...405..273W,Kasen_2010,Metzger_2011}.

Despite rotation being recognized as an important parameter controlling the evolution of massive stars \cite{Maeder_2000}, little is known about the rotation rates of the inner cores of massive stars nearing CC. %Although rapid rotation leaves an imprint in the gravitational wave spectrum produced at core bounce (Ott et al. 2012, Abdikamalov et al. 2014, Klion et al. 2015, Fuller et al. 2015), a gravitational wave detection of a galactic supernovae is likely decades away.
The best observational constraints stem from measurements of the rotation rates of the compact remnants following CC. For instance, a few low-mass black hole X-ray binary systems have been measured to have large spins that can only be accounted for by high spins at birth \citep{Axelsson_2011,Miller_2011,Wong_2012}. However, the rotation rates of young NSs show little evidence for rapid rotation ($P \lesssim 10 \, {\rm ms}$) at birth. The most rapidly rotating young pulsars include PSR J0537-6910 ($P=16\,{\rm ms}$) and the Crab pulsar ($P=33\,{\rm ms}$), whose birth periods have been estimated to be $P_i \lesssim 10\,{\rm ms}$ \citep{marshall:98} and $P_i \sim 19 \, {\rm ms}$ \citep{kaspi:02}, respectively. Many young NSs appear to rotate much more slowly, with typical periods of hundreds of ms \citep{lai:96,gotthelf:13,2015arXiv150107220D}. In general, pulsar observations seem to indicate a broad range of initial birth periods in the vicinity of tens to hundreds of milliseconds \citep{faucher:06,popov:10,gullon:14}. Hence, rapidly rotating young NSs appear to be the exception rather than the rule. 


Theoretical efforts have struggled to produce slow rotation rates. In the absence of strong AM transport mechanisms within the massive star progenitor, NSs would invariably be born rotating near break-up \cite{heger:00}. \citet{Heger_2005} and \citet{Suijs_2008} examined the effect of magnetic torques generated via the Tayler-Spruit (TS) dynamo \citep{spruit:02}, and found typical NS spin periods at birth (assuming AM conservation during core-collapse and the ensuing supernova) of several milliseconds. \citet{wheeler:14} implemented magnetic torques due to MRI and the TS dynamo, and were able to reach iron core rotation rates of $P_{c} \sim 500 \, {\rm s}$, corresponding to NS spin periods of $P \sim 25 \, {\rm ms}$. These efforts are promising, but the operation of both mechanisms within stars has been debated \citep[e.g.][]{Zahn_2007}, and theoretical uncertainties abound.


Recent asteroseismic advances have allowed for the measurement of core rotation rates in low-mass red giant stars (\citealt{beck:12,beck:14,mosser:12,deheuvels:12,deheuvels:14}). In these stars, the core rotates much faster than the surface and one cannot assume nearly rigid rotation as suggested in \citet{spruit:98}. However, the cores of low-mass red giants rotate much slower than can be explained via hydrodynamic AM transport mechanisms or magnetic torques via the TS dynamo \citep{cantiello:14}. If similar angular momentum transport mechanisms operate in more massive objects, this suggests that the pre-collapse cores of massive stars may rotate slower than predicted by many previous theoretical investigations.


Internal gravity waves (IGW) constitute a powerful energy and AM transport mechanism in stellar interiors. Several studies (\citealt{kumar:97,zahn:97,kumar:99,talon:02,talon:03,talon:05,talon:08,charbonnel:05,denissenkov:08,fullerwave:14}, hereafter F14) have found that convectively generated IGW can redistribute large quantities of AM within low-mass stars. IGW may partially account for the rigid rotation of the Sun's radiative interior and the slow rotation of red giant cores, although magnetic torques are also likely to be important (\citealt{denissenkov:08}, F14). IGW may also be important in more massive stars, and \cite{lee:14} found that convectively generated IGW in Be-type stars may instigate outbursts that expel mass into the decretion disk.


Convectively excited IGW may also have a strong influence on the evoution of massive stars nearing CC. Indeed, after core carbon exhaustion, waves are the most effective energy transport mechanism within radiative zones, as photons are essentially frozen in and neutrinos freely stream out. In two recent papers, \citet{quataert:12} and \citet{shiode:14} (hereafter QS12 and SQ14) showed that the prodigious power carried by convectively excited waves (on the order of $10^{10} L_\odot$ during Si burning) can sometimes unbind a large amount of mass near the stellar surface, and may substantially alter the pre-collapse stellar structure. IGW are ubiquitously seen in simulations of late burning stages \citep{meakin:06,meakina:07,meakinb:07}, although existing simulations have not quantified their long-term impact. 


In this paper, we examine AM transport due to convectively excited IGW within massive stars, focusing primarily on AM transport during late burning stages (He, C, O, and Si burning). We find that IGW are generally capable of redistributing large amounts of AM before CC despite the short stellar evolution time scales. IGW emitted from convective shells propagate into the radiative core and may be able to substantially slow its rotation. In the limit of very efficient core spin-down via magnetic torques, we show that stochastic influxes of AM via IGW set a minimum core rotation rate which is comparable with the broad distribution of low rotation rates ($P \lesssim 500 \,{\rm ms}$) observed for most young NSs.


Our paper is organized as follows. In Section 2, we describe our massive star models, the generation of IGW during various stages of stellar evolution, and the AM they transport. In Section 3, we investigate whether the IGW can spin down the cores of massive stars, attempting to determine a minimum core rotation period. In Section 4, we consider whether IGW can stochastically spin up a very slowly rotating core, attempting to determine a maximum core rotation period. In Section 5, we conclude with a discussion of our results and their implications for core collapse, supernovae, and the birth of compact objects. 
