During each shell burning phase, the convective shell emits roughly $N$ wave packets, with 
\begin{equation}
\label{eqn:Nwave}
N \sim \frac{\omega_{\rm c} T_{\rm shell}}{2 \pi}
\end{equation}
where $T_{\rm shell}$ is the length of the shell burning phase. For Si shell burning, $N \sim 400$. This relatively small number demonstrates that the stochastic nature of the wave emission process can be important during late burning phases. The total core AM in each direction exhibits a random walk and thus has a Gaussian distribution centered around zero and with standard deviation 
\begin{equation}
\label{eqn:Jmean}
\sigma_{J} = \sqrt{ \frac{N}{3} } J_w.
\end{equation}
The magnitude of the total AM vector then has a Maxwellian distribution, with standard deviation $\sigma_J$. The corresponding spin frequency (assuming internal torques eventually restore rigid rotation in the core) also has a Maxwellian distribution, given by 
\begin{equation}
\label{eqn:jmaxwell}
f(\Omega) = \sqrt{\frac{2}{\pi}} \frac{\Omega^2}{\sigma_\Omega^3} e^{- \Omega^2/(2 \sigma_\Omega^2)} \,.
\end{equation}
where $\sigma_\Omega = \sigma_J/I_c$, and $I_c$ is the moment of inertia of the radiative core. A little algebra yields the corresponding spin period distribution,
\begin{equation}
\label{eqn:pdist}
f(P) = \sqrt{\frac{2}{\pi}} \frac{\sigma_P^3}{P^4} e^{- \sigma_P^2/(2 P^2)} \,,
\end{equation}
with $\sigma_P = 2 \pi/\sigma_\Omega$.


The expected angular spin frequency corresponding to equation \ref{eqn:jmaxwell} is
\begin{equation}
\label{eqn:omex}
\Omega_{\rm ex} = \sqrt{\frac{4 \omega_c T_{\rm shell} }{3 \pi^2}} \frac{J_w}{I_c} \, .
\end{equation}
The expected spin frequency scales as $\Omega_{\rm ex} \propto \omega_c^{-1/2} L_c T_{\rm shell}^{1/2}$. Thus, more energetic and long-lived convection yields higher expected rotation rates. The short duration $T_{\rm shell}$ of later burning phases largely counteracts their increased vigor, and we find that earlier burning phases are generally capable of depositing more AM.

The stochastic spin-up process described above will only occur under certain conditions. First, as already mentioned, the core and burning shell must be slowly rotating, or else the stochastic spin-up will have a negligible effect. Second, all waves (both prograde and retrograde) must be absorbed by the core. In the cores of massive stars, this is likely to occur because of non-linear breaking due to geometric focusing as waves approach the center of the star (see Appendix \ref{wavestar}). Third, stochastic spin-up can only proceed as long as $\Omega_{\rm ex} < \omega_c$. If $\Omega_{\rm ex}$ approaches $\omega_c$, wave filtering processes as described in Section \ref{igw} will alter the subsequent dynamics. Most of our estimates below have $\Omega_{\rm ex} \ll \omega_c$, so we believe they are valid estimates of minimum spin rates. For C-burning in the optimistic wave flux estimate ($L_{\rm wave} \propto \mathcal{M}^{5/8}$), however, $\Omega_{\rm ex}$ approaches $\omega_c$, so this value of $\Omega_{\rm ex}$ lies near the maximum rotation rate achievable through stochastic spin-up for our stellar model.

Moreover, stochastic spin-up can only occur if other sources of AM transport (e.g., magnetic torques) operate on longer time scales. This could be the case during late burning phases when magnetic torques become ineffective (\citealt{Heger_2005,wheeler:14}). We can also estimate a minimum magnetic coupling time between core and envelope via the Alven wave crossing time $t_A \approx r_c \sqrt{\rho_c}/B$, with $B$ the approximate magnetic field strength. Typical neutron star field strengths of a few times $10^{12} \, {\rm G}$ imply field strengths of $\sim 10^8 \,{\rm G}$ in the iron core, which yields $t_A \sim 5 \times 10^4 \,{\rm s}$, much longer than the Si shell burning time (see Table 1). Although magnetic torques may suppress stochastic spin-up during He/C/O burning phases, we expect them to have a negligible impact during Si burning.

It is also possible that IGWs will generate significant amounts of shear within the core, as IGW tend to amplify small amounts of shear \cite{Fuller_2014}. In the absence of strong magnetic torques, such shear amplification may generate shear layer oscillations \citep{Kumar_1999,Talon_2002} or differential rotation that encompasses much of the radiative core \cite{Denissenkov_2008}. The timescale for shear generation of amplitude $\omega_c$ is $t_{\rm shear} = I_c \omega_c/\dot{J}$, and we find $t_{\rm shear} \sim 2 \times 10^{4} \, {\rm s}$  for Si shell burning (using the conservative estimate from equation \ref{eqn:Ewaves}). We conclude that IGW may generate significant shear within the $T_{\rm shell} \sim 7\times 10^{3} \, {\rm s}$ Si shell burning life time, although not enough to encompass the entire core and prevent further influx of IGWs. The stochastic spin-up process can still proceed as we have outlined, although the distribution of AM within the core at collapse is unclear, and significant differential rotation is possible. 

Figure \ref{fig:MassiveIGWspin} shows the distribution in maximum spin period of the pre-collapse iron core, $P_{\rm max,Fe}$, assuming the spin of the core is determined by stochastic IGW spin-up. We have plotted the values of $P_{\rm max,Fe}$, if the core spin rate is set during C, O, or Si burning. We have also plotted the corresponding spin rate of the $M_{\rm NS} \sim 1.4 M_\odot$, $R_{\rm NS} \sim 12 \, {\rm km}$ NS, with $I_{\rm NS} = 0.25 M_{\rm NS} R_{\rm NS}^2$, if its AM is conserved during the CC SN. We find that C, O, and Si burning generate maximum iron core rotation periods in the range of tens to thousands of seconds, depending on the burning phase and efficiency of IGW generation. Si burning most plausibly sets the pre-SN conditions because it is the last convective burning phase before CC and is least likely to be affected by magnetic torques. We find stochastic spin-up during Si burning leads to $300 \, {\rm s} \lesssim P_{\rm max,Fe} \lesssim  10^4 \, {\rm s}$. The corresponding NS rotation rate is $20\,{\rm ms} \lesssim P_{\rm NS} \lesssim 400 \, {\rm ms}$. Hence, we find that very slow core rotation rates, as suggested by \cite{Spruit_1998}, are unlikely. Nor do we expect that there is a population of NSs born with very long spin periods, $P \gtrsim 2 \, {\rm s}$, at least from progenitors with ZAMS mass $10 \, M_\odot < M < 20 \, M_\odot$. 

The distribution of NS spin periods shown in Figure \ref{fig:MassiveIGWspin} appears broadly consistent with those inferred for young NSs \citep{faucher:06,popov:10,gullon:14}. We are therefore tempted to speculate that the stochastic wave flux scenario described above may be the dominant process setting the spin rates of newly born NSs. If so, this scenario predicts that the rotation rate and direction of the NS is uncorrelated with the rotation of the envelope of the progenitor star, in contrast to any sort of magnetic spindown mechanism. However, there are several caveats to keep in mind. First, the scenario presented above can only proceed if the core is initially very slowly rotating, which requires efficient magnetic/wave core spin down to occur before Si burning. Second, the NS rotation rate may be changed during the supernova, by fallback effects, or by the r-mode instability (\citealt{Andersson_1998,Andersson_1999}, see \citealt{ott:2009} for a review). Finally, there is a considerable amount of uncertainty in the wave flux and frequency spectrum. Since the minimum core rotation rate set by stochastic waves is proportional to the wave energy flux (which is uncertain at an order of magnitude level), there is an equal amount of uncertainty in the induced rotation rates. 
