\section{Introduction}
\label{sec:Introduction}

Massive, young stars are sites of nucleosynthesis, not just of the stable nuclides, but radioactive isotopes as well.  Long-lived radioisotopes with decay times of billions of years or longer, such as $^{40}$K, mix into the interstellar medium (ISM), accumulating and provide a very low level source of energetic radiation in all gas \citep{Cameron62,Umebayashi81}.  More unstable isotopes are also synthesized; some have decay times of a few years or less and cannot reach most of the ISM.  In between those two extremes are the short lived radionuclides (SLRs) of the ISM: with $\sim \textrm{Myr}$ decay times, they can be present in a galaxy's gas but only as long as star formation replenishes their abundances.  The most prominent of the SLRs is \textrm{$^{26}$Al}, which is detected in gamma-ray decay lines from star-formation regions throughout the Milky Way \citep{Mahoney84,Diehl06}.  Another SLR that has recently been detected in gamma-ray lines is $^{60}$Fe \citep{Harris05}.

SLRs are not just passive inhabitants of the ISM.  By releasing energetic particles and radiation when they decay, they inject power that can heat or ionize the surrounding matter.  In the Milky Way's molecular clouds, the radioactivity is overwhelmed by that of cosmic rays, which sustain an ionization rate of $\zeta_{\rm CR} \approx 5 \times 10^{-17}\ \sec^{-1}$.  Rapidly star-forming galaxies known as starbursts may have elevated levels of cosmic rays and ionization rates a thousand times or more higher than the Milky Way (\citealt*{Suchkov93}; \citealt{Papadopoulos10-CRDRs}).  However, it is possible that cosmic rays are stopped by the abundant gas in starbursts before they enter the densest molecular gas.  Gamma rays can provide ionization through large columns; but while the gamma-ray ionization rate can reach up to $\zeta_{\gamma} \approx 10^{-16}\ \sec^{-1}$ in the densest starbursts, in most starbursts gamma rays sustain relatively weak ionization \citep{Lacki12-GRDRs}.  SLRs like \textrm{$^{26}$Al} can in principle provide ionization through any column of gas, and if abundant enough, maintain moderate levels of ionization.  The major open question for SLR-induced ionization is how well mixed the SLRs are with the gas of the galaxy, a process which takes anything up to 100 Myr; if the mixing times are longer than a few Myr, the SLRs are not abundant in most star-forming cores \citep{Meyer00,Huss09}.

Meteorites recording the composition of the primordial Solar system demonstrate that SLRs were present during its formation. Assuming the SLRs were not created in situ by energetic radiation from the Sun \citep{Lee98}, the SLRs provide evidence that the Solar system formed near a star-forming region with young massive stars \citep[e.g.,][]{Adams10}.  In fact, \textrm{$^{26}$Al} was overabundant by a factor $\sim 6$ in the primordial Solar system, with $X (\textrm{$^{26}$Al}) \approx 10^{-10}$ ($\textrm{$^{26}$Al} / ^{27}{\rm Al} \approx 5 \times 10^{-5}$), compared to its present day abundances in the Milky Way (e.g., \citealt*{Lee77}; \citealt*{MacPherson95}; \citealt{Diehl06}; \citealt{Huss09}).  Their quick decay time also indicate the Solar system formed quickly, within about a few Myr.  SLRs, particularly \textrm{$^{26}$Al}, were a primary source of ionization in the Solar Nebula \citep{Stepinski92,Finocchi97,Umebayashi09}, affecting the conductivity and ultimately accretion rate in the protoplanetary disc.  Moreover, \textrm{$^{26}$Al} and other SLRs may have regulated the early geological evolution of the Solar system by being a major source of heat in early planetesimals, driving their differentiation and rock metamorphism \citep[e.g.,][]{Hutcheon89,Grimm93,Shukolyukov93}.

The contemporary Milky Way, with a star-formation rate (SFR) of a few stars per year is not a typical environment for most of the star-formation in the history of the Universe, however.  Roughly 5-20\% of star-formation at all times occurred in rapid starbursts mainly driven by galaxy-galaxy interactions and mergers \citep{Rodighiero11,Sargent12}.  Furthermore, most of the star-formation in `normal' galaxies occurred in massive galaxies with a much higher star-formation rate ($\ga 10\ \textrm{M}_{\sun}\ \textrm{yr}^{-1}$) at redshifts $z$ of 1 and higher, when most star-formation took place \citep[e.g.,][]{Magnelli09}.  These high star-formation rates translate into large masses of SLRs present in these galaxies.  I { will} show that \textrm{$^{26}$Al} in these galaxies, if it is well mixed with the gas, can sustain rather high ionization rates in their ISMs.  This has consequences for both star formation and planet formation.

When necessary, I assume a Hubble constant of $H_0 = 72\ \textrm{km}~\textrm{s}^{-1}\ \textrm{Mpc}^{-1}$, a matter density of $\Omega_M = 0.25$, and a cosmological constant $\Omega_{\Lambda} = 0.75$ for the cosmology.

\section{The Equilibrium Abundance of SLR\lowercase{s}}
In a one-zone model of a galaxy, which disregards spatial inhomogeneities, the complete equation for the SLR mass $M_{\rm SLR}$ in the ISM is
\begin{equation}
\label{eqn:MeqSLRFull}
\frac{dM_{\rm SLR}}{dt} = Q_{\rm SLR}(t) - \frac{M_{\rm SLR} (t)}{\tau_{\rm SLR}},
\end{equation}
where $\tau_{\rm SLR}$ is the lifetime of the SLR in the galaxy.  $Q_{\rm SLR} (t)$, the injection rate of the SLR, depends on the past star-formation history:
\begin{equation}
\label{eqn:QSLR}
Q_{\rm SLR} (t) = \int_{-\infty}^t Y_{\rm SLR}(t - t^{\prime}) \times {\rm SFR}(t^{\prime}) dt^{\prime}.
\end{equation}
For a coeval stellar population of age $t$, the yield $Y_{\rm SLR}(t)$ is the mass ejection rate of the SLR into the interstellar medium per unit stellar mass \citep{Cervino00}.

If there are no big fluctuations in the star-formation rate over the past few Myr, then the SLR abundance approaches a steady-state.  The equilibrium mass of a SLR in a galaxy is proportional to its star-formation (or supernova) rate averaged over the previous few Myr.  We can then { parametrize} the injection of SLRs in the ISM by a yield $\Upsilon_{\rm SLR}$ per supernova:
\begin{equation}
\Upsilon_{\rm SLR} = \varepsilon \int_0^{\infty} Y(t^{\prime\prime}) dt^{\prime\prime},
\end{equation}
regardless of whether SNe are actually the source of SLRs.  The $\varepsilon$ factor is the ratio of the supernova rate $\Gamma_{\rm SN}$ and star-formation rate.  Then the equilibrium SLR mass is given by \citep[e.g.,][]{Diehl06}
\begin{equation}
\label{eqn:MeqSLR}
M_{\rm SLR}^{\rm eq} = \Gamma_{\rm SN} \Upsilon_{\rm SLR} \tau_{\rm SLR}.
\end{equation}
The supernova rate is proportional to the star-formation rate, so $\Gamma_{\rm SN} = \varepsilon {\rm SFR}$.  The abundance of an SLR is given by $X_{\rm SLR} = M_{\rm SLR}^{\rm eq} m_H / (M_{\rm gas} m_{\rm SLR})$, where $m_{\rm SLR}$ is the mass of one atom of the SLR and $M_{\rm gas}$ is the gas mass in the galaxy.  Therefore the abundance of an SLR is
\begin{equation}
X_{\rm SLR} = \varepsilon \frac{\rm SFR}{M_{\rm gas}} \frac{\Upsilon_{\rm SLR} \tau_{\rm SLR} m_H}{m_{\rm SLR}}
\end{equation}
The quantity $M_{\rm gas} / {\rm SFR} = \tau_{\rm gas}$ is the gas consumption time.  Note that it is related to the specific star formation rate, ${\rm SSFR} = M_{\star} / {\rm SFR}$, as $\tau_{\rm gas} = f_{\rm gas} / ((1 - f_{\rm gas}) {\rm SSFR})$, where $M_{\star}$ is the stellar mass and $f_{\rm gas} = M_{\rm gas} / (M_{\rm gas} + M_{\star})$ is the gas fraction.  Therefore, we can express the equilibrium mass of the SLR in a galaxy as
\begin{equation}
X_{\rm SLR} = \frac{\varepsilon \Upsilon_{\rm SLR} \tau_{\rm SLR} m_H}{\tau_{\rm gas} m_{\rm SLR}} = \varepsilon \frac{1 - f_{\rm gas}}{f_{\rm gas}} {\rm SSFR} \frac{\Upsilon_{\rm SLR} \tau_{\rm SLR} m_H}{m_{\rm SLR}}
\end{equation}
Finally, the ratio of SLR abundance in a galaxy to that in the Milky Way is
\begin{eqnarray}
\nonumber \frac{X_{\rm SLR}}{X_{\rm SLR}^{\rm MW}} & = & \frac{\tau_{\rm gas}^{\rm MW}}{\tau_{\rm gas}} \frac{\tau_{\rm SLR}}{\tau_{\rm SLR}^{\rm MW}}\\
& = & \frac{1 - f_{\rm gas}}{1 - f_{\rm gas}^{\rm MW}} \frac{f_{\rm gas}^{\rm MW}}{f_{\rm gas}} \frac{\rm SSFR}{\rm SSFR^{\rm MW}} \frac{\tau}{\tau^{\rm MW}},
\end{eqnarray}
with a MW superscript referring to values in the present day Milky Way.  Thus, galaxies with short gas consumption times (and generally those with high SSFRs) should have high abundances of SLRs.  The reason is that in such galaxies, more of the gas is converted into stars and SLRs within the residence time of an SLR.

The greatest uncertainty in these abundances is the residence time $\tau_{\rm SLR}$.  In the Milky Way, these times are just the radioactive decay times, defined here as the e-folding time.  In starburst galaxies, however, much of the volume is occupied by a hot, low density gas which forms into a galactic wind with characteristic speeds $v$ of several hundred { kilometres} per second (e.g., \citealt{Chevalier85}; \citealt*{Heckman90}; \citealt{Strickland09}).  If massive stars emit SLRs at random locations in the starburst, most of them will dump their SLRs into the wind phase of the ISM.  The wind-crossing time is $\tau_{\rm wind} = 330\ \textrm{kyr}\ (h / 100\ \textrm{pc}) (v / 300\ \textrm{km}~\textrm{s}^{-1})^{-1}$, where $h$ is the gas scale-height.  The equilibrium time in starburst galaxies is then $\tau = [\tau_{\rm decay}^{-1} + \tau_{\rm wind}^{-1}]^{-1}$.  Furthermore, the SLRs ejected into the wind may never mix with the molecular gas, so the fraction of SLRs injected into the molecular medium may be $\ll 1$ (I discuss this issue further in section~\ref{sec:Mixing}).  However, very massive stars are found close to their birth environments where there is a lot of molecular gas to enrich, and these may be the source of \textrm{$^{26}$Al}, as supported by the correlation of the 1.809 MeV \textrm{$^{26}$Al} decay line emission and free-free emission from massive young stars \citep{Knoedlseder99}.

Turning to the specific example of \textrm{$^{26}$Al}, I note that the yield of \textrm{$^{26}$Al} is thought to be $\Upsilon_{\rm Al-26} \approx 1.4 \times 10^{-4}\ \textrm{M}_{\sun}$ per supernova \citep{Diehl06}.  For a Salpeter initial mass function from $0.1 - 100\ \textrm{M}_{\sun}$, the supernova rate is $\Gamma_{\rm SN} = 0.0064\ \textrm{yr}^{-1} ({\rm SFR} / \textrm{M}_{\sun}\ \textrm{yr}^{-1})$, or $\Gamma_{\rm SN} = 0.11\ \textrm{yr}^{-1} (L_{\rm TIR} / 10^{11}\ \textrm{L}_{\sun})$ in terms of the total infrared ($8 - 1000\ \mu\textrm{m}$) luminosity $L_{\rm TIR}$ of starbursts (\citealt{Kennicutt98}; \citealt*{Thompson07}).  If I suppose all of the \textrm{$^{26}$Al} is retained by the molecular gas, so that the residence time is the \textrm{$^{26}$Al} decay time of 1.04 Myr, then the equilibrium abundance of \textrm{$^{26}$Al} in a galaxy is just
\begin{equation}
\label{eqn:XAl26Numer}
X (\textrm{$^{26}$Al}) = 3.4 \times 10^{-11} \left(\frac{\tau_{\rm gas}}{\textrm{Gyr}}\right)^{-1} = 1.7 \times 10^{-9} \left(\frac{\tau_{\rm gas}}{20\ \textrm{Myr}}\right)^{-1}
\end{equation}

\subsection{High-Redshift Normal Galaxies}
\label{sec:MSGalaxies}

The star-formation rates and stellar masses of normal star-forming galaxies lie on a `main sequence' with a characteristic SSFR that varies weakly, if at all, with stellar mass \citep[e.g.,][]{Brinchmann04}.  However, the characteristic SSFR evolves rapidly with redshift \citep[e.g.,][]{Daddi07,Noeske07,Karim11}, with ${\rm SSFR} \propto (1 + z)^{2.8}$ out to $z \approx 2.5$ -- a rise of factor $\sim 30$ \citep{Sargent12}.  At $z \ga 2.5$, the SSFR of the main sequence then seems to remain constant \citep{Gonzalez10}.

Countering this rise in the SSFR, the gas fractions of normal galaxies at high $z$ were also higher: the high equilibrium masses of SLRs are diluted to some extent by higher gas masses.  \citet{Hopkins10} provide a convenient equation, motivated by the Schmidt law \citep{Kennicutt98}, to describe the evolution of gas fraction:
\begin{equation}
\label{eqn:fGas}
f_{\rm gas} (z) = f_0 [1 - (t_L (z) / t_0) (1 - f_0^{3/2})]^{-2/3},
\end{equation}
assuming a gas fraction $f_0$ at $z = 0$, with a look back time of $t_L (z) = \int_0^z dz^{\prime} / [H_0 (1 + z^{\prime}) \sqrt{\Omega_{\Lambda} + \Omega_M (1 + z^{\prime})^3}]$ and a current cosmic age of $t_0$ \citep[see also][]{Hopkins09}.  Since the gas fractions of normal galaxies at present are small, the evolution at low redshifts can be approximated as $f_{\rm gas} (z) = f_0 [1 - (t_L (z) / t_0)]^{-2/3}$.

After calculating the mean abundances of SLRs in normal galaxies, I find that the rapid SSFR evolution overwhelms the modest evolution in $f_{\rm gas}$ at high $z$: the SLR abundances of normal galaxies evolves quickly.  These enhancements are plotted in Fig.~\ref{fig:NormGalaxy}.

Observational studies of high redshift main sequence galaxies indicate a slower evolution in $\tau_{\rm gas}$, resulting from a quicker evolution of $f_{\rm gas}$.  Although equation~\ref{eqn:fGas} implies that $f_{\rm gas}$ was about twice as high ten billion years ago at $z \approx 2$, massive disc galaxies are observed with gas fractions of $\sim 40$ -- $50\%$, which is 3 to 10 times greater than at present \citep[e.g.,][]{Tacconi10,Daddi10}.  According to \citet{Genzel10}, the typical (molecular) gas consumption time at redshifts 1 to 2.5 was $\sim 500\ \textrm{Myr}$.  In the \citet{Daddi10} sample of BzK galaxies at $z \approx 1.5$, gas consumption times are likewise $\sim 300$ -- $700\ \textrm{Myr}$.  To compare, the molecular gas consumption times at the present are estimated to be 1.5 to 3 Gyr \citep{Diehl06,Genzel10,Bigiel11,Rahman12}, implying an enhancement of a factor 3 to 6 in SLR abundances at $z \ga 1$.  But note that the BzK galaxies are not the direct ancestors of galaxies like the present Milky Way, which are less massive.  The SSFR, when observed to have any mass dependence, is greater in low mass galaxies at all $z$ \citep{Sargent12}.  This means that lower mass galaxies at all $z$ have shorter $\tau_{\rm gas}$, as indicated by observations of present galaxies \citep{Saintonge11}.  The early Milky Way therefore may have had a gas consumption time smaller than $500\ \textrm{Myr}$.

So far, I have ignored possible metallicity $Z$ dependencies in the yield $\Upsilon$ of SLRs.  It may be generally expected that star-forming galaxies had lower metallicity in the past, since less of the gas has been processed by stellar nucleosynthesis.  However, observations of the age-metallicity relation of G dwarfs near the Sun reveal that they have nearly the same metallicity at ages approaching 10 Gyr (e.g., \citealt{Twarog80,Haywood06}; \citealt*{Holmberg07}), though the real significance of the lack of a trend remains unclear, since there is a wide scatter in metallicity with age \citep[see the discussion by][]{Prantzos09}.  Observations of external star-forming galaxies find weak evolution at constant stellar mass, with metallicity $Z$ decreasing by $\sim 0.1-0.2$ dex per unit redshift (\citealt*{Lilly03}; \citealt{Kobulnicky04}).

After adopting a metallicity dependence of $Z(z) = Z(0) \times 10^{-0.2 z}$ (0.2 dex decrease per unit redshift), I show in the revised SLR abundances Fig.~\ref{fig:NormGalaxy} assuming that the SLR yield goes as $Z^{-1}$, $Z^{-0.5}$, $Z^{0.5}$, $Z$, $Z^{1.5}$, and $Z^2$.  If the yields are smaller at lower metallicity, the SLR abundances are still elevated at high redshift, though by not as much for metallicity-independent yields.  As an example, the yield of \textrm{$^{26}$Al} in the winds of Wolf-Rayet stars is believed to scale as $\Upsilon \propto Z^{1.5}$ \citep{Palacios05}.  According to \citet{Limongi06}, these stellar winds contribute only a minority of the \textrm{$^{26}$Al} yield, so it is unclear how the \textrm{$^{26}$Al} yield really scales.  \citet{Martin10} considered the \textrm{$^{26}$Al} and \textrm{$^{60}$Fe} yields from stars with half Solar metallicity.  They found that, because reduced metallicity lowers wind losses, more SLRs are produced in supernovae.  This mostly compensates for the reduced wind \textrm{$^{26}$Al} yield, and actually raises the synthesized amount of \textrm{$^{60}$Fe}.

\begin{figure}
\centerline{\includegraphics[width=8cm]{f1.eps}}
\caption{Plot of the SLR abundance enhancements in normal galaxies lying on the `main sequence', for a gas fraction evolution described by equation~\ref{eqn:fGas}.  The rapid evolution of SSFRs leads to big enhancements of SLRs at high $z$.  Even during the epoch of Solar system formation, the mean SLR abundance was twice the present value.  The different lines are for different $f_{\rm gas}$ at $z = 0$, assuming SLR yields are independent of metallicity: 0.05 (dotted), 0.1 (solid), 0.2 (dashed).  The shading shows the abundances for $0.05 \le f_{\rm gas} \le 0.2$ when the SLR yield depends on metallicity, assuming a 0.2 dex decrease in metallicity per unit redshift. \label{fig:NormGalaxy}}
\end{figure}

\subsection{Starbursts}

\begin{table*}
\begin{minipage}{170mm}
\caption{$^{26}$A\lowercase{l} Abundances and Associated Ionization Rates}
\label{table:Al26Abundances}
\begin{tabular}{lccccccccc}
\hline
Starburst                             & SFR & $\Gamma_{\rm SN}$ & $M_{\rm Al-26}^{\rm eq}$ & $M_H$ & $\tau_{\rm gas}$ & $X (\textrm{$^{26}$Al})^a$ & $\displaystyle \frac{^{26}\rm Al}{^{27}\rm Al}^b$ & $\zeta_{\rm Al-26}(e^+)^c$ & $\zeta_{\rm Al-26}(e^+ \gamma)^d$\\
                                      & ($\textrm{M}_{\sun}\ \textrm{yr}^{-1}$) & (yr$^{-1}$) & ($\textrm{M}_{\sun}$) & ($\textrm{M}_{\sun}$) & ($\textrm{Myr}$) & & & $(\sec^{-1})$ & $(\sec^{-1})$\\
\hline
Milky Way ($z = 0$)$^e$       & 3.0   & 0.019                & 2.8   & $4.5 \times 10^9$ & 1500 & $2.4 \times 10^{-11}$ & $9.4 \times 10^{-6}$ & $1.9 \times 10^{-20}$ & $7.1 \times 10^{-20}$\\
Galactic Centre CMZ$^f$       & 0.071 & $4.6 \times 10^{-4}$ & 0.067 & $3 \times 10^7$   & 420  & $8.6 \times 10^{-11}$ & $3.4 \times 10^{-5}$ & $6.9 \times 10^{-20}$ & $2.6 \times 10^{-19}$\\
NGC 253 core$^{g,h}$          & 3.6   & 0.023                & 3.3   & $3 \times 10^7$   & 8.3  & $4.3 \times 10^{-9}$  & $1.8 \times 10^{-3}$ & $3.4 \times 10^{-18}$ & $1.3 \times 10^{-17}$\\
M82$^{g,i}$                   & 10.5  & 0.067                & 9.8   & $2 \times 10^8$   & 19   & $1.9 \times 10^{-9}$  & $7.5 \times 10^{-4}$ & $1.5 \times 10^{-18}$ & $5.7 \times 10^{-18}$\\
Arp 220 nuclei$^j$            & 50    & 0.3                  & 44    & $10^9$            & 20   & $1.7 \times 10^{-9}$  & $6.7 \times 10^{-4}$ & $1.3 \times 10^{-18}$ & $5.0 \times 10^{-18}$\\
Submillimeter galaxy$^k$      & 1000  & 6.4                  & 930   & $2.5 \times 10^{10}$ & 25   & $1.4 \times 10^{-9}$ & $5.7 \times 10^{-4}$ & $1.1 \times 10^{-18}$ & $4.3 \times 10^{-18}$\\
BzK galaxies$^l$              & 200   & 1                    & 200   & $7 \times 10^{10}$   & 400  & $1 \times 10^{-10}$  & $4 \times 10^{-5}$  & $8 \times 10^{-20}$ & $3 \times 10^{-19}$\\
\hline
\end{tabular}
\\$^a$: Mean abundance of \textrm{$^{26}$Al}, calculated assuming the \textrm{$^{26}$Al} is well-mixed with the gas and resides there for a full decay time (instead of, for example, a wind-crossing time).
\\$^b$: Calculated assuming Solar metallicity with $\log_{10} [N(^{27}{\rm Al})/N(H)] = -5.6$.
\\$^c$: Ionization rate from \textrm{$^{26}$Al} with the derived abundance, with ionization only from MeV positrons released by the decay, assuming effective stopping.
\\$^d$: Ionization rate from \textrm{$^{26}$Al} with the derived abundance, where ionization from the 1.809 MeV decay line and 0.511 keV positron annihilation gamma rays is included, assuming they are all stopped.
\\$^e$: Supernova rate and gas mass from \citet{Diehl06}; SFR calculated from supernova rate using Salpeter IMF for consistency.
\\$^f$: Inner 100 pc of the Milky Way.  SFR and $\Gamma_{\rm SN}$ from IR luminosity in \citet*{Launhardt02}; gas mass from \citet{Molinari11}.  \citet{PiercePrice00} gives a gas mass of $5 \times 10^7\ \textrm{M}_{\sun}$.
\\$^g$: SFR and $\Gamma_{\rm SN}$ from IR luminosity in \citet{Sanders03}.
\\$^h$: Gas mass from \citet*{Harrison99}.
\\$^i$: Gas mass from \citet{Weiss01}.
\\$^j$: Assumes IR luminosity of $3 \times 10^{11}\ \textrm{L}_{\sun}$ for SFR and $\Gamma_{\rm SN}$ and gas mass given in \citet{Downes98}.
\\$^k$: Typical gas mass and SFR of { submillimetre} galaxies from \citet{Tacconi06}.
\\$^l$: Mean SFR and gas mass of the 6 BzK galaxies in \citet{Daddi10}, which are representative of main sequence galaxies at $z \approx 1.5$.
\end{minipage}
\end{table*}

The true starbursts, driven by galaxy mergers and galaxies interacting with each other, represent about $\sim 10\%$ of star formation at all redshifts \citep{Rodighiero11,Sargent12}.  They have SSFRs that are up to an order of magnitude higher than $z = 2$ normal galaxies.  The mean, background abundances of SLRs in starbursts are therefore about 100 times greater than the present day Milky Way.

I show the \textrm{$^{26}$Al} abundances in some nearby starburst galaxies in Table~\ref{table:Al26Abundances}.  In the Galactic Centre region, the \textrm{$^{26}$Al} abundance is only twice that of the present Milky Way as a whole.  However, the \textrm{$^{26}$Al} abundances are extremely high in the other starbursts, $\sim 2 \times 10^{-9}$, about twenty times that of the primordial Solar nebula.  The $^{26}{\rm Al}/^{27}{\rm Al}$ ratio in these starbursts is also very high.  Assuming Solar metallicity with an $^{27}$Al abundance of $\log_{10} [N(^{27}{\rm Al})/N(H)] = -5.6$ \citep{Diehl06}, this ratio is $\sim (0.6 - 1.8) \times 10^{-3}$.  Again, this ratio for Solar metallicity gas is $\sim 10 - 30$ times higher than that of the early Solar Nebula, $\sim 5 \times 10^{-5}$.

\section{Systematic Uncertainties}
\subsection{Effects of Variable Star-Formation Rates}

The steady-state abundance (equation~\ref{eqn:MeqSLR}) is only appropriate when the star-formation rate is slowly varying on time-scales of a few Myr.  Since young stellar populations produce SLRs for several Myr, and since \textrm{$^{26}$Al} and \textrm{$^{60}$Fe} themselves survive for $\ga 1\ \textrm{Myr}$, the injection rate of SLRs is smoothed over those time-scales (equation~\ref{eqn:QSLR}).  Very high frequency fluctuations in the SFR therefore have little effect on the abundance of SLRs.  In the opposite extreme, when the fluctuations in SLRs are slow compared to a few Myr, we can simply take the present SFR and use it in equation~\ref{eqn:MeqSLR} for accurate results.  However, intermediate frequency variability invalidates the use of equation~\ref{eqn:MeqSLR}, and can result in the SLR abundance being out of phase with the SFR.

Normal main sequence galaxies at high redshift built up their stellar populations over Gyr times, evolving secularly \citep[c.f.,][]{Wuyts11}.  They are also large enough to contain many stellar clusters, so that stochastic effects average out.  It is reasonable to suppose that they have roughly constant SFRs over the relevant time-scales.  True starbursts, on the other hand, are violent events that last no more than $\sim 100\ \textrm{Myr}$, as evinced by their short $\tau_{\rm gas}$.  They are relatively small, so stochastic fluctuations in their star-formation rates are more likely.  \citet{Forster03} studied the nearby, bright starburst M82 and concluded that its star-formation history is in fact bursty.  The star-formation histories of other starbursts are poorly known, but \citet{Mayya04} present evidence for large fluctuations on $\sim 4\ \textrm{Myr}$ times.

I estimate the magnitude of these fluctuations for the prototypical starburst M82 with the full equation for SLR mass in a one-zone model (equation~\ref{eqn:MeqSLRFull}).  The solution to equation~\ref{eqn:MeqSLRFull} for $M_{\rm SLR}$ is
\begin{equation}
M_{\rm SLR} (t) = \int_{-\infty}^t {\rm SFR}(t^{\prime}) \times m_{\rm SLR}(t^{\prime}) dt^{\prime},
\end{equation}
where
\begin{equation}
m_{\rm SLR}(t^{\prime}) = \int_{-t^{\prime}}^0 Y_{\rm SLR}(t^{\prime\prime}) \exp\left(-\frac{t^{\prime} - t^{\prime\prime}}{\tau_{\rm SLR}}\right) dt^{\prime\prime}.
\end{equation}
The quantity $m_{\rm SLR}(t^{\prime})$ represents the SLR mass in the ISM from a coeval stellar population of unit mass and age $t^{\prime}$.  It is given by \citet{Cervino00} and \citet{Voss09} for \textrm{$^{26}$Al} and $^{60}$Fe.

I use the star-formation history derived by \citet{Forster03} for the `3D region' of M82, which consists of two peaks at 4.7 Myr ago and 8.9 Myr ago.  The peaks are modelled as Gaussians with the widths given in \citet{Forster03} (standard deviations $\sigma$ of 0.561 Myr for the more recent burst, and 0.867 Myr for the earlier burst).  I convert the star-formation rate from a Salpeter IMF from 1 to 100$\ \textrm{M}_{\sun}$ given in \citet{Forster03} to a Salpeter IMF from 0.1 to 100$\ \textrm{M}_{\sun}$ for consistency with the rest of the paper.\footnote{I ignore the relatively small difference between the upper mass limit of 100$\ \textrm{M}_{\sun}$ in \citet{Forster03} and 120$\ \textrm{M}_{\sun}$ in \citet{Cervino00} and \citet{Voss09}.  Since stars with masses 100 to 120$\ \textrm{M}_{\sun}$ can affect stellar diagnostics, converting to that IMF may require an adjustment to the star-formation history beyond a simple mass scaling.}  This region does not include the entire starburst; it has roughly $1/3$ of the luminosity of the starburst, but the stellar mass formed within the 3D region over the past 10 Myr gives an average SFR of 10 $\textrm{M}_{\sun}\ \textrm{yr}^{-1}$ in the \citet{Forster03} history.  Note that \citet*{RodriguezMerino11} derives a different age distributions for stellar clusters (compare with \citealt{Satyapal97}).  \citet{Strickland09} has also argued that the star-formation history of M82's starburst core is not well constrained before 10 Myr ago (as observed from Earth), and may have extended as far back as 60 Myr ago.  Thus, I take the \citet{Forster03} history merely as a representative example of fluctuating SFRs.

\begin{figure}
\centerline{\includegraphics[width=8cm]{f2.eps}}
\caption{History of the SLR masses in M82's `3D region' ISM for the star-formation history given in \citet{Forster03}.  The black lines are for \textrm{$^{26}$Al} and grey lines are for $^{60}$Fe; solid lines are using the yields in \citet{Voss09} and dashed lines are using the \citet{Cervino00} yields.  We presently observe M82 at $t = 0$; I assume there are no bursts of star-formation after then, so that the masses inevitably decay away. \label{fig:M82SLRHistory}}
\end{figure}

The calculated \textrm{$^{26}$Al} (black) and \textrm{$^{60}$Fe} (grey) masses are plotted in Fig.~\ref{fig:M82SLRHistory}.  At first, there is no SLR mass in the starburst, because it takes a few Myr for SLR injection to start.  With the \citet{Voss09} yields, the SLR masses rise quickly and peak $\sim 5\ \textrm{Myr}$ ago (as observed from Earth).  The SLR masses drop afterwards.  Yet they are still within a factor of 1.7 of their peak values even now, $\sim 5\ \textrm{Myr}$ after the last star-formation burst.  If there is no further star-formation, the SLRs will mostly vanish over the next 10 Myr.  The \citet{Cervino00} yields predict a greater role for supernovae from lower mass stars, so the fluctuations are not as great; the \textrm{$^{60}$Fe} mass remains roughly the same even 10 Myr from now.  As long as there has been recent star-formation in the past $\sim 5\ \textrm{Myr}$, the SLR abundances are at least half of those predicted by the steady-state assumption.

There is a more fundamental reason to expect that the steady-state SLR abundances are roughly correct for starbursts.  A common way of estimating star-formation rates in starbursts is to use the total infrared luminosity \citep{Kennicutt98}, which is nearly the bolometric luminosity for these dust-obscured galaxies.  Young stellar populations, containing very massive stars, are brighter and contribute disproportionately to the bolometric luminosity.  Therefore, both the luminosity and the SLR abundances primarily trace young stars.  To compare the bolometric luminosity, I ran a Starburst99 (v6.04) model of a $Z = 0.02$ metallicity coeval stellar population with a Salpeter IMF ($dN/dM \propto M^{-2.35}$) between 0.1 and 120$\ \textrm{M}_{\sun}$ \citet{Leitherer99}.  I then calculate the SFR that would be derived from these { luminosities} using the \citet{Kennicutt98} conversion, and then from that, the expected steady-state SLR masses from equation~\ref{eqn:MeqSLR}.  The `bolometric' \textrm{$^{26}$Al} masses are compared to the actual masses in Fig.~\ref{fig:LBolVsMAl26}.

\begin{figure}
\centerline{\includegraphics[width=8cm]{f3.eps}}
\caption{How the bolometric luminosity traces \textrm{$^{26}$Al} mass for a coeval stellar population with age $t$.  The grey line is the predicted steady state \textrm{$^{26}$Al} mass I would predict from the bolometric luminosity of the population, whereas the black lines are the actual mass of \textrm{$^{26}$Al} (solid for \citealt{Voss09} and dashed for \citealt{Cervino00}).\label{fig:LBolVsMAl26}}
\end{figure}

Although the very youngest stellar populations are bright but not yet making SLRs, the bolometric luminosity (grey) is a good tracer of \textrm{$^{26}$Al} mass (black) for stellar populations with ages between 3 and 20 Myr.  For most of the interval, the bolometric \textrm{$^{26}$Al} masses are within a factor 2 of the actual masses.  For populations between 15 Myr and 20 Myr, the \citet{Voss09} and \citet{Cervino00} predictions envelop the bolometric \textrm{$^{26}$Al} masses.  At 20 to 25 Myr old, the bolometric \textrm{$^{26}$Al} masses are about twice the true masses.  For older populations still, the true \textrm{$^{26}$Al} masses finally die away while the bolometric luminosity only slowly declines.  Note that, if stars have been forming continuously for the past 100 Myr, over half of the luminosity comes from stars younger than 20 Myr.  Thus, the use of the bolometric luminosities introduces a factor $\la 3$ systematic error.

In short, the use of bolometric luminosity as a SFR indicator, and the natural variability in the star-formation rates of starbursts can lead to overestimations of the SLR abundances by a factor $\sim 3$.  But I estimate the SLR abundances of true starbursts are a hundred times higher than in the present Milky Way (equation~\ref{eqn:XAl26Numer} and Table~\ref{table:Al26Abundances}).  The ratio is so great that the systematic effects do not undermine the basic conclusion that SLR abundances are much larger in true starbursts.

\subsection{Are SLRs mixed quickly enough into the gas?}
\label{sec:Mixing}
Although the average levels of SLRs in starbursts and high-$z$ normal galaxies are high, that does not by itself mean the SLRs influence the environments for star-formation.  While SLRs can play an important role in star-forming regions, by elevating the ionization rates and by being incorporated into solid bodies, SLRs trapped in ionized gas are irrelevant for these processes.

The mixing of metals from young stars into the ISM gas mass is usually thought to be very slow in the present Milky Way, compared to SLR lifetimes.  The massive stars responsible for making SLRs often live in star clusters, which blow hot and rarefied bubbles in the ISM.  Supernovae also excavate the coronal phase of the ISM \citep{McKee77}.  Turbulence within the bubbles mixes the SLRs and homogenizes their abundances \citep[e.g.,][]{Martin10}, over a time scale $t_{\rm mix} \approx L / v_{\rm turb}$, where $L$ is the outer scale of turbulence (typical size of the largest eddies) and $v_{\rm turb}$ is the turbulent speed \citep{Roy95,Pan10}.  The large outer scale of turbulence, $\sim$100 -- 1000$\ \textrm{pc}$, and the slow turbulent speeds ($\sim 5$--$10\ \textrm{km}~\textrm{s}^{-1}$) in the Milky Way imply mixing times of $\sim 10$ -- $200\ \textrm{Myr}$.  Even if the \textrm{$^{26}$Al} is homogenized within the superbubbles, this low density hot gas requires a long time to mechanically affect cold star-forming clouds, because of the large density contrast \citep{deAvillez02}.  Mixing between the phases, particularly warm and hot gas, is accelerated by Rayleigh-Taylor and Kelvin-Helmholtz instabilities \citep{Roy95}, but overall, mixing takes tens of Myr to operate in the Milky Way (\citealt{deAvillez02}; see also \citealt{Clayton83}, where mixing times are between the warm ISM from evaporated H I clouds, cool and large H I clouds, and molecular clouds).  Thus, SLRs like \textrm{$^{26}$Al} are thought to decay long before they are mixed thoroughly with the star-forming gas.  Indeed, studies of the abundances of longer lived isotopes in the primordial Solar system supports longer mixing times of $\sim 50$ -- 100$\ \textrm{Myr}$ \citep[e.g.,][]{Meyer00,Huss09}.

It is thought that these obstacles existed, at least qualitatively, in the $z \approx 0.45$ Milky Way, when the Solar system formed.  These problems are part of the motivation for invoking a local source of SLRs, including energetic particles from the Sun itself \citep{Lee98}, injection from a nearby AGB star (\citealt*{Busso03}), or injection from an anomalously nearby supernova \citep{Cameron77} or Wolf-Rayet star (\citealt*{Arnould97}).  Recently, though, several authors proposed models that might overcome the mixing obstacle, where young stars are able to inject SLRs into star-forming clouds.  A motivation behind these models is the idea that molecular clouds are actually intermittent high density turbulent fluctuations in the ISM \citep[e.g.,][]{MacLow04}, and the supernovae that partly drive the turbulence -- indirectly forming the molecular clouds -- also are the sources of SLRs \citep{Gounelle09}.   In the model of \citet{Gounelle09}, old superbubbles surrounding stellar clusters form into molecular clouds after { ploughing} through the ISM for $\sim$10 Myr.  Supernovae continue going off in the star clusters, adding their SLRs into these newborn molecular clouds \citep[see also][]{Gounelle12}.  Simulations by \citet*{Vasileiadis13} also demonstrate that SLRs from supernovae going off very near giant molecular clouds are mixed thoroughly with the star-forming gas.  On a different note, \citet{Pan12} argued that supernovae remnants are clumpy, and that clumps could penetrate into molecular clouds surround star clusters and inject SLRs.

If these scenarios are common, then a large fraction of the produced SLRs reaches the star-forming gas before decaying.  In fact, these mechanisms may be so efficient that SLRs are concentrated only into star-forming molecular gas, a minority of the Milky Way gas mass.  If so, then the abundance of SLRs within Galactic molecular gas ($M_{\rm SLR} / M_{\rm H2}$) is greater than the mean background level ($M_{\rm SLR} / M_{\rm gas}$); in this way, SLR abundances could reach the elevated levels that existed in the early Solar system \citep{Gounelle09,Vasileiadis13}.

{ On the other hand, young stars can trigger star-formation in nearby gas without polluting them.  This can occur when a shock from a supernova or from an overpressured H II region propagates into a molecular cloud, causing the cores within it to collapse (\citealt{Bertoldi89}; \citealt*{Elmegreen95}).  This process has been inferred to happen in the Upper Scorpius OB association \citep{Preibisch99}.  Since the cores are pre-existing, they may not be enriched with SLRs (although supernova shocks can also inject SLRs into a molecular cloud; see \citealt{Boss10} and \citealt{Boss12}).  The triggering can also occur before any supernovae enrich the material with SLRs.  Star formation can also be triggered when a shock from a H II region sweeps up a shell of material, which eventually becomes gravitationally unstable and collapses (\citealt{Elmegreen77}; see also \citealt{Gritschneder09}).}

I note, however, that the homogeneity of the \textrm{$^{26}$Al} abundance in the early Solar system is controversial; if the abundance was inhomogeneous, that is inconsistent with efficient SLR mixing within the Solar system's progenitor molecular cloud.  Although \citet*{Villeneuve09} conclude that \textrm{$^{26}$Al} had a constant abundance, \citet{Makide11} find that the \textrm{$^{26}$Al} abundance varied during the earliest stages of Solar system formation, when { aluminium} first condensed from the Solar nebula.

What is even less clear, though, is how similar the Milky Way is to starbursts and the massive high-$z$ normal galaxies that host the majority of the cosmic star formation.  As in the Milky Way, supernovae in starbursts like M82 probably blast out a hot phase.  But the hot phase escapes in a rapid hot wind in starbursts with high star-formation densities ($\ga 0.1\ \textrm{M}_{\sun}\ \textrm{yr}^{-1}\ \textrm{kpc}^{-2}$; \citealt{Chevalier85,Heckman90}).  There is direct evidence for this hot wind from X-ray observations \citep{Strickland07,Strickland09}.  Furthermore, supernova remnants are observed to expand quickly in M82, implying that they are going off in a very low density environment (e.g., \citealt{Beswick06}; compare with \citealt{Chevalier01}).  Cool and warm gas is observed outflowing alongside the hotter wind, possibly entrained by the hot wind \citep{Strickland09}.  Whereas the edges of superbubbles eventually cool and fade back into the ISM in the Milky Way after a 10 -- 100 Myr, any superbubble gas that does cool in these starbursts could still be pushed out by the wind.  If the SLRs are trapped in the wind, they reside in these starbursts for only a few hundred kyr.

But the ISM conditions in the starbursts are much different, with higher densities in all phases, higher pressures, and higher turbulent speeds \citep[e.g.,][]{Smith06}.  Starbursts are physically small, with radii of $\sim 200\ \textrm{pc}$ at $z = 0$ -- comparable to the size of some individual superbubbles in the Milky Way.  The eddy sizes therefore must be smaller and mixing processes could be faster.  To demonstrate just how small superbubbles are in starbursts, \citet*{Silich07} modelled the superbubble surrounding the star cluster M82 A-1 in M82, which has a mass of $2 \times 10^6\ \textrm{M}_{\sun}$.  They find that the wind propagates only for a few parsecs before being shocked and cooled.  Stellar ejecta in the core of the cluster also cool rapidly in their model.  The turbulent mixing time is therefore much smaller, $\sim 200\ \textrm{kyr}$ for an eddy length of 10 pc and a turbulent speed of $50\ \textrm{km}~\textrm{s}^{-1}$.

Conditions are even more extreme in present day compact Ultraluminous Infrared Galaxy starbursts, where the ISM is so dense ($\sim 10^4\ \textrm{cm}^{-3}$; \citealt{Downes98}) that a hot phase may be unable to form (\citealt{Thornton98}; \citealt*{Thompson05}).  Instead, the ISM is almost entirely molecular \citep{Downes98}.  Indeed, observations of supernovae in Arp 220 indicate they are going off in average density molecular gas \citep{Batejat11}.  Supernovae remnants fade within a few tens of kyr into the ISM, due to powerful radiative losses \citep{McKee77}.  The SLRs then are incorporated into the molecular ISM in a turbulent mixing time, the whole process taking just a few hundred kyr.  The main uncertainty is then, not whether the SLRs are injected into the molecular gas, but whether these SLR-polluted regions of the molecular gas fill the entire starburst.  Turbulent mixing smooths abundances over regions the size of the largest eddies \citep{Pan10}, but if the distribution of SLR injection sites varies over larger scales, the final SLR abundance may also vary on these large scales.

We know very little about the conditions in high redshift galaxies.  Star formation in main sequence galaxies is dominated by massive galaxies with large star-formation rates at high redshift.  These massive galaxies are several kpc in radius, but contain large amounts of molecular gas \citep{Daddi10}.  They also have large star-formation densities and host winds.  In the more extreme galaxies, radiative bremsstrahlung losses stall any hot wind \citep{Silich10}.  Turbulent speeds in these galaxies are high \citep{Green10}, implying faster turbulent mixing than in the Milky Way.  But it is not clear which phase the SLRs are injected into or how long it takes for them to mix throughout star formation regions.  The effects of clustering in the ISM is also uncertain, but it probably is important in these galaxies, where huge clumps ($\ga 10^8\ \textrm{M}_{\sun}$ and a kpc wide) are observed \citep[e.g.,][]{Genzel11}.

To summarize, while there are reasons to expect that most of the SLRs synthesized by young stars in the Milky Way decay before reaching star-forming gas, this is not necessarily true in starbursts or high-$z$ normal galaxies.  Turbulent mixing is probably fast, at least in compact starbursts which are physically small.  On the other hand, winds might blow out SLRs before they reach the star-forming gas, at least in the weaker starbursts.  Clearly, this issue deserves further study.

\section{Implications}
\subsection{Implications for the early Solar system and Galactic stars of similar age}
The rapid evolution in SSFRs implies that Galactic background SLR abundances were up to twice as high during the epoch of Solar system formation (4.56 Gyr ago; $z \approx 0.44$).  If the evolution of the Milky Way's gas fraction was comparable to that in observed massive main sequence galaxies, the enhancement may have been only $\sim 50\%$ above present values (see the discussion in section~\ref{sec:MSGalaxies}; \citealt{Genzel10}).  The inferred primordial abundances of $^{60}$Fe in the Solar system are in fact up to twice as high as in the contemporary Milky Way, as determined with gamma-ray lines \citep{Williams08,Gounelle08}.  This overabundance is cited as evidence for an individual, rare event enriching the early Solar system, or the gas that formed into the Solar system, with SLRs.  However, my calculations show this is not necessarily the case: the twice high abundances of $^{60}$Fe in the early Solar system can arise \emph{simply because the Galaxy was more efficient at converting gas into stars 4.5 Gyr ago.}

The primordial abundance of \textrm{$^{26}$Al} in the Solar system was about six times higher than the mean Galactic value at present \citep{Diehl06}, or three times higher than the mean Galactic abundance at $z = 0.44$ assuming that equation~\ref{eqn:fGas} holds.  Even so, the normal effects of galaxy evolution are a potential contributor to the greater \textrm{$^{26}$Al} abundances, assuming efficient mixing of \textrm{$^{26}$Al} with the molecular gas of the Milky Way.  Furthermore, the high abundances of \textrm{$^{26}$Al} in the early Solar system are actually typical of star-formation at $z \approx 1 - 2$ -- when most cosmic star-formation occurred.  In this sense, the early Solar system's \textrm{$^{26}$Al} abundance may be normal for most planetary systems in the Universe.

As I have discussed in Section~\ref{sec:Mixing}, it is not clear whether the background abundances of \textrm{$^{26}$Al} and other SLRs actually represent those of typical star-forming gas; if mixing takes more than a few Myr, these SLRs could not have affected star and planet formation \citep{Meyer00,deAvillez02,Huss09}.  But although there may have been a wide distribution of abundances if mixing is inefficient, the mean of the distribution is still higher simply because there were more supernovae and young stars per unit gas mass.  Thus, a greater fraction of star formation occurred above any given threshold in the past.  In addition, the Galactic background level can be meaningful if a large fraction of the SLRs from young stars make it into the cold gas, and \citet{Gounelle09}{,} \citet{Gounelle12}, \citet{Pan12}, and \citet{Vasileiadis13} have presented mechanisms where this can happen.  Although there is suggestive evidence that these mechanisms did not operate for the Solar system \citep{Makide11}, there is no reason they could not have worked for other star systems of similar ages.  Then the Solar system's relatively high abundance of SLRs, and \textrm{$^{60}$Fe} in particular, may be common for Galactic stars of similar ages, even if through a different process.

Finally, as I noted, these conclusions depend on how the yields of SLRs from massive stars change with metallicity, and what the mean Galactic metallicity was at the epoch of Solar system formation.

\subsection{The Ionization Rate and Physical Conditions in Starbursts' Dense Clouds}
Radioactive decay from SLRs injects energy in the form of daughter particles into the ISM.  The decay particles, with typical energies of order an MeV, ionize gas and ultimately heat the ISM, if they do not escape entirely.  The high abundances of SLRs, including \textrm{$^{26}$Al}, can alter the ionization state of molecular gas in these galaxies.  The ionization rate, in particular, is important in determining whether magnetic fields can thread through the densest gas.  I focus here on the contribution from \textrm{$^{26}$Al}, which dominated the ionization rate from SLRs in the primordial Solar system \citep{Umebayashi09}.  For the sake of discussion, I assume that the SLRs are well-mixed into the gas, despite the uncertainties (section~\ref{sec:Mixing}).

Each \textrm{$^{26}$Al} decay releases an energy $E_{\rm decay}$ into the ISM in the form of MeV positrons and gamma rays.  If each atom in the ISM takes an energy $E_{\rm ion}$ to be ionized, each \textrm{$^{26}$Al} decay can therefore ionize $E_{\rm decay} / E_{\rm ion}$ atoms.  In 82\% of the decays, a positron with kinetic energy of 1.16 MeV is released, and the positron is slowed by ionization losses \citep{Finocchi97}.  The minimum energy per decay, after accounting for this branching ratio, that goes into ionization is $E_e^{\rm min} = 0.95\ \textrm{MeV}$, when the medium stops the positron (inflight annihilation losses are negligible at these energies; \citealt{Beacom06}).  The annihilation of the positron with { an} electron in the ISM produces gamma rays of total energy $2 \times 0.511 = 1.022$ MeV.  In addition, in very nearly all \textrm{$^{26}$Al} decays, a 1.809 MeV gamma ray is produced.  These gamma rays only interact with the ISM over columns of several $\textrm{g}~\textrm{cm}^{-2}$, so only in particularly dense regions will they contribute to the ionization \citep{Finocchi97}.  When they do, $E_{\rm decay}^{e \gamma} = 3.60\ \textrm{MeV}$.  \citet{Stepinski92} gives $E_{\rm ion}$ as 36.3 eV, so that the ionization rate is
\begin{equation}
\zeta_{\rm Al-26} = \frac{X_{\rm Al-26} E_{\rm decay}}{(36.3\ \textrm{eV}) \tau_{\rm decay}}.
\end{equation}

In terms of gas consumption time, the ionization rate from \textrm{$^{26}$Al} is
\begin{equation}
\zeta = (1.4 - 5.1) \times 10^{-18}\ \sec^{-1}\ \left(\frac{\tau_{\rm gas}}{20\ \textrm{Myr}}\right)^{-1}.
\end{equation}
My results for the mean ionization rate from \textrm{$^{26}$Al} of some characteristic starbursts are shown in Table~\ref{table:Al26Abundances}; they are in the range $10^{-18} - 10^{-17}\ \sec^{-1}$.  The maximal ionization rates are roughly an order of magnitude higher than that found in early Solar system, a dense environment with $\zeta \approx (0.6 - 1) \times 10^{-18}\ \sec^{-1}$ \citep{Finocchi97,Umebayashi09}.

Even if the \textrm{$^{26}$Al} is in the cold star-forming gas, it could actually be condensed into dust grains instead of existing in the gas phase.  Yet the decay products still escape into the ISM from within the grain.  The attenuation of gamma rays at $\sim 1$ MeV is dominated by Compton scattering, requiring columns of a few $\textrm{g}~\textrm{cm}^{-2}$ to be important.  Thus, gamma rays pass freely through dust grains that are smaller than a { centimetre}.  The energy loss rate of relativistic electrons or positrons in neutral matter is approximately
\begin{equation}
\frac{dK}{ds} = \frac{9}{4} m_e c^2 \sigma_T \sum_j \frac{\rho_j Z_j}{A_j m_H} \left[\ln \frac{K + m_e c^2}{m_e c^2} + \frac{2}{3} \ln \frac{m_e c^2}{\mean{E_j}}\right]
\end{equation}
from \citet{Schlickeiser02}, where $K$ is the particle kinetic energy, $s$ is the physical length, and $\sigma_T$ is the Thomson cross section.  The sum is over elements $j$; for heavy elements $Z_j \approx A_j / 2$, $\rho_j$ is the partial density of each element within the grain, and $\mean{E_j}$ is related to the atomic properties of the element.  I take the bracketed term to have a value $\sim 5$ and $\rho \approx 3\ \textrm{g}~\textrm{cm}^{-2}$.  Then the stopping length is $K / (dK / ds) \approx 0.3\ \textrm{cm}\ (K / \textrm{MeV})$, much bigger than the typical grain radius.  Thus, \textrm{$^{26}$Al} (and other SLRs) in dust grains still contribute to the ionization of the ISM.

On the other hand, are the positrons actually stopped in starburst molecular clouds, or do they escape?  For neutral interstellar gas, the stopping column of MeV positrons is $\sim K \rho / (dK / ds) \approx 0.2\ \textrm{g}~\textrm{cm}^{-2}$ { through ionization and excitation of atoms.}  \citep{Schlickeiser02}.  { In cold molecular gas, ionization and excitation continue to cool the positrons until $K \approx 10\ \textrm{eV}$, at which point they start annihilating by charge exchange reactions or they thermalize \citep*{Guessoum05}.}  The column densities of starbursts range from $\sim 0.1 - 10\ \textrm{g}~\textrm{cm}^{-2}$ \citep[e.g.,][]{Kennicutt98}, and the columns through individual molecular clouds are expected to be similar to those of the galaxies (\citealt*{Hopkins12}).  In the denser molecular clouds, positrons are stopped even if they are not confined at all.  In massive main sequence galaxies, the columns are $\sim 0.1\ \textrm{g}~\textrm{cm}^{-2}$ \citep{Daddi10}, insufficient to stop positrons moving in straight lines.  If magnetic waves with a wavelength near the positron gyroradius scale exist in these clouds, they efficiently scatter positrons and confine them.  { As a result, the propagation of the positrons can be described with a diffusion constant, as widely used when interpreting the Galactic GeV positron spectrum (e.g., \citealt{Moskalenko98}; \citealt*{Hooper09}), although it is unclear how relevant these studies are for MeV positrons \citep{Jean09,Prantzos11,Martin12}.} However, these waves are probably damped quickly in dense neutral gas (see { \citealt{Jean09}; \citealt*{Higdon09};} \citealt{Prantzos11} and references therein).  On the other hand, positrons move along magnetic field lines, and if the lines themselves are twisted on a scale $\lambda_B$, the positrons are forced to random walk with a similar mean free path { \citep{Martin12}}.  As long as $\lambda_B$ is less than about a third of the molecular cloud size, then positrons are stopped in these galaxies.

If it is well mixed with the molecular gas, does \textrm{$^{26}$Al} dominate the ionization rate in molecular gas in starbursts, and if so, what physical conditions does it induce?  The starburst \textrm{$^{26}$Al} ionization rates are about an order of magnitude lower than the canonical cosmic ray-sustained ionization rates in most Milky Way molecular clouds, but in some of the densest Galactic starless cores, the ionization rate drops to $\sim 10^{-18}\ \sec^{-1}$ \citep{Caselli02,Bergin07}.  Cosmic rays in starbursts can sustain much higher ionization rates (up to $\sim 10^{-14}\ \sec^{-1}$; c.f., \citealt{Papadopoulos10-CRDRs}), but cosmic rays can be deflected by magnetic fields, possibly preventing them from penetrating high columns.  { Aside from cosmic rays themselves, starbursts are also bright sources of GeV gamma rays, which are generated when cosmic rays interact with the prevalent gas \citep[e.g.,][]{Ackermann12}.  These gamma rays can penetrate molecular clouds and induce low levels of ionization \citep{Lacki12-GRDRs}.}  In \citet{Lacki12-GRDRs}, I found that the gamma-ray ionization rate in starbursts can be anywhere from $10^{-22} - 10^{-16}\ \sec^{-1}$, with values of $\sim (1 - 3) \times 10^{-19}\ \sec^{-1}$ in M82 and $\sim (5 - 8) \times 10^{-17}\ \sec^{-1}$ in Arp 220's radio nuclei.  In the dense clouds of most starbursts, \textrm{$^{26}$Al} radioactivity could exceed the ionization rate over gamma rays, setting a floor to the ionization rate.  In the most extreme starbursts, with mean gas surface densities of $\ga 3\ \textrm{g}~\textrm{cm}^{-2}$ (c.f. equation 10 of \citealt{Lacki12-GRDRs}), however, gamma-ray ionization is more important, since the gamma-ray ionization rate depends strongly on the density of gas and compactness of the starbursts.  Unlike the uncertainty of how SLRs and their positron decay products are transported and mixed with the gas of starbursts, gamma rays propagate relatively simply, so the gamma ray ionization rates are more secure.

An \textrm{$^{26}$Al}-dominated ionization rate has implications for the physical conditions of star-forming clouds.  According to \citet{McKee89}, the ionization fraction of a cloud with hydrogen number density $n_H$ is
\begin{equation}
\label{eqn:xELow}
x_e \approx 1.4 \times 10^{-8} \left(\frac{\zeta}{10^{-18}\ \sec^{-1}}\right)^{1/2} \left(\frac{n_H}{10^4\ \textrm{cm}^{-3}}\right)^{-1/2}
\end{equation}
when the ionization rate is low.  We see that the ionization fraction of cloud with density $n_H = 10^4\ \textrm{cm}^{-3}$ { is} $(1 - 4) \times 10^{-8}$, if the ionization is powered solely by \textrm{$^{26}$Al} decay.  For these ionization fractions, the ambipolar diffusion time of a molecular core, the time for magnetic fields to slip from the gas, is a few times its free-fall time.  Since clouds with strong magnetic fields do not collapse until the field slips away by ambipolar diffusion \citep{Mestel56,Mouschovias76}, this means that \textrm{$^{26}$Al}-ionized clouds in starbursts collapse quasi-statically, as in the Milky Way.

On the other hand, the energy injection from \textrm{$^{26}$Al} has essentially no effect on the gas temperature. \citet{Papadopoulos10-CRDRs} gives the minimum gas temperature of gas as:
\begin{equation}
T_k^{\rm min} = 6.3\ \textrm{K}\ [(0.0707 n_4^{1/2} \zeta_{-18} + 0.186^2 n_4^3)^{1/2} - 0.186 n_4^{3/2}]^{2/3},
\end{equation}
which was derived under the assumption that there is no heating from interactions with dust grains or the dissipation of turbulence in the gas, for gas with density $n_4 = (n_H / 10^4\ \textrm{cm}^{-3})$ and ionization rate $\zeta_{-18} = (\zeta / 10^{-18}\ \sec^{-1})$.  In typical starbursts, I find that \textrm{$^{26}$Al} decay alone heats gas of density $n_H = 10^{4}\ \textrm{cm}^{-3}$ to $\sim 2 - 5\ \textrm{K}$ ($0.1 - 0.5\ \textrm{K}$ for $n_H = 10^6\ \textrm{cm}^{-3}$).  As I note in \citet{Lacki12-GRDRs}, under such conditions, dust heating is more likely to set the temperature of the gas than ionization, raising the temperature to the dust temperature for densities $\ga 40000\ \textrm{cm}^{-3}$.

\section{Conclusions}

The high SSFRs of starbursts and high-$z$ normal galaxies implies high abundances of \textrm{$^{26}$Al} and other SLRs in their ISMs.  In true starbursts, these abundances are enormous, with $X (\textrm{$^{26}$Al}) \approx 10^{-9}$ and $^{26}$Al/$^{27}$Al $\approx 10^{-3}$.  The SSFRs of normal galaxies evolve rapidly with $z$; even taking into account higher gas fractions, the SLR abundances were about 3 -- 10 times higher at $z \approx 2$ than in the present Milky Way.  Even at the epoch of Solar system formation, the mean SLR abundance of the Milky Way was 1.5 to 2 times as high as at the present (Fig.~\ref{fig:NormGalaxy}).  This alone could explain the high abundances of $^{60}$Fe in the early Solar system, and reduce the discrepancy in the \textrm{$^{26}$Al} abundances from a factor of $\sim 6$ to $\sim 3$.  In this way, the cosmic evolution of star-forming galaxies may have direct implications for the early geological history of the Solar system.  The first main uncertainty is whether the SLRs produced by massive stars is well-mixed with the molecular gas: they may instead be ejected by the galactic winds known to be present in starbursts and high-$z$ galaxies, or decay before they can propagate far from its injection site, or before it penetrates cold gas.  I discussed these uncertainties in section~\ref{sec:Mixing}.  The other uncertainty is how SLR yields depend on metallicity.

The most direct way to test the high \textrm{$^{26}$Al} abundances of starburst galaxies is to detect the 1.809 MeV gamma-ray line produced by the decay of \textrm{$^{26}$Al}, directly informing us of the equilibrium mass of \textrm{$^{26}$Al}.  Whether most of the \textrm{$^{26}$Al} is ejected by the superwind can be resolved with spectral information.  The turbulent velocities of molecular gas in starbursts, $\sim 100\ \textrm{km}~\textrm{s}^{-1}$ \citep{Downes98}, is much smaller than the bulk speed of the superwind, which is hundreds or even thousands of $\textrm{km}~\textrm{s}^{-1}$ \citep{Chevalier85}.  Unfortunately, the \textrm{$^{26}$Al} line fluxes predicted for even the nearest external starbursts ($\sim 10^{-8}\ \textrm{cm}^{-2}\ \sec^{-1}$) are too low to detect with any planned instrument (\citealt*{Lacki12-MeV}).  However, \citet{Crocker11-Wild} have argued that the inner 100 pc of the Galactic Centre are an { analogue} of starburst galaxies, launching a strong superwind.  The \textrm{$^{26}$Al} line from this region should have a flux of $\sim 2 \times 10^{-5}\ \textrm{cm}^{-2}\ \sec^{-1}$, easily achievable with possible future MeV telescopes like Advanced Compton Telescope (ACT; \citealt{Boggs06}) or Gamma-Ray Burst Investigation via Polarimetry and Spectroscopy (GRIPS; \citealt{Greiner11}).  A search for the \textrm{$^{26}$Al} signal from this region may inform us on its propagation, since it is nearby and resolved spatially{.}\footnote{\citet{Naya96} reported that the \textrm{$^{26}$Al} signal from the inner Galaxy had line widths corresponding to speeds of several hundred $\textrm{km}~\textrm{s}^{-1}$, but this was not verified by later observations { by RHESSI} \citep{Smith03} { and INTEGRAL} \citep{Diehl06-Linewidth}.  { Instead, recent observations indicate that Galactic \textrm{$^{26}$Al} is swept up in superbubbles expanding at 200 $\textrm{km}~\textrm{s}^{-1}$ into the interarm regions \citep{Kretschmer13}.} However, this signal is from the entire inner Galactic disc on kiloparsec scales; the inner 100 pc of the Galactic Centre is a much smaller region and just a small part of this signal, so the kinematics of its \textrm{$^{26}$Al} are currently unconstrained.  Current observations of the \textrm{$^{26}$Al} decay signal from the Galactic Centre region are summarized in \citet{Wang09}.}  If the \textrm{$^{26}$Al} generated in the Centre region is actually advected away by the wind, the `missing' \textrm{$^{26}$Al} should be visible several hundred pc above and below the Galactic Plane near the Galactic Centre{.}\footnote{This will also be true in other starbursts, but these starbursts would not be resolved spatially by proposed MeV telescopes.  The total amount of \textrm{$^{26}$Al} line emission from other starbursts would therefore not inform us of whether it is in the starburst proper or in the superwind; other information, such as the Doppler width of the line, is necessary to determine that.}  Resolved measurements of the Galactic Centre can also inform us on whether the \textrm{$^{26}$Al} is present in all molecular clouds in the region (and therefore is well-mixed), or just if it is trapped near a few injection sites.

If the SLRs do mix with the star-forming molecular gas of these galaxies, there are implications for both their star formation and planet formation.  Ionization rates of $10^{-18} - 10^{-17}\ \sec^{-1}$, like those in some Milky Way starless cores, result from the energy injection of \textrm{$^{26}$Al} decay in starbursts.  While cosmic ray ionization rates can easily exceed those ionization rates by orders of magnitude in gas they penetrate into, and while gamma rays produce higher ionization rates in the most extreme starbursts like Arp 220, \textrm{$^{26}$Al} might dominate the ionization rate in the dense clouds shielded from cosmic rays in typical starbursts.  In starbursts' protoplanetary discs, \textrm{$^{26}$Al} can provide moderate ionization through all columns, possibly eliminating the `dead zones' where there is little accretion (e.g., \citealt{Gammie96}; \citealt*{Fatuzzo06}).  Any planetesimals that do form in starbursts may have much higher radiogenic heating from \textrm{$^{26}$Al}.  Admittedly, studying the geological history of planets, much less planetesimals, in other galaxies is very difficult for the forseeable future.  However, at $z \approx 2$ the Milky Way likely had background SLR abundances $\sim 10$ times higher than at present, so the effects of elevated SLR abundances may be studied in planetary systems around old Galactic stars.

On a final point, \citet{Gilmour09} propose that the elevated abundances of \textrm{$^{26}$Al} in the early Solar system are mandated by anthropic selection, since high SLR abundances are necessary for planetesimal differentiation and the loss of volatiles, but that explanation may be difficult to maintain.  If high \textrm{$^{26}$Al} abundances, far from being very rare, are actually typical of high-$z$ and starburst solar systems (and indeed, much of the star-formation in the Universe's history), the anthropic principle would imply that most technological civilizations would develop in solar systems formed in these environments.  Instead of asking why we find ourselves in a system with an \textrm{$^{26}$Al} abundance just right to power differentiation and evaporate volatiles, we must ask why we find ourselves in one of the rare solar systems with sufficient \textrm{$^{26}$Al} that formed in a normal spiral galaxy at $z \approx 0.4$, instead of the common \textrm{$^{26}$Al}-enriched solar systems formed at $z \approx 2$ or in starbursts.

\section*{Acknowledgements}
During this work, I was supported by a Jansky Fellowship from the National Radio Astronomy Observatory.  The National Radio Astronomy Observatory is operated by Associated Universities, Inc., under cooperative agreement with the National Science Foundation.


\section{BEAM-BEAM EFFECTS}

At the Interaction Point (IP) of the ILC, beam-beam effects due to the strong
electromagnetic fields that the bunches experience during collisions cause a
mutual focusing, called pinch effect, which enhances the luminosity in the case
of \ep collisions. The opposite is true for \ee collisions. In this case the
luminosity is reduced by mutual defocusing, or anti-pinching and is only about
20$\%$ of the \ep one (see Fig.~\ref{lumi-GP}). Moreover this repulsion between
the bunches causes the luminosity to drop with a vertical offset at the IP much
more rapidly for the \ee case than for \ep. Another effect of this strong
repulsive electromagnetic field is the much steeper beam-beam deflection curve
(see Fig.~\ref{deflection-GP}). Since the fast intra-train feedback system used
to maintain the beams in collision at the IP \cite{feedback-GWhite} exploits
these deflections as its main signal and because of the higher sensitivity to
the vertical offsets, it is important to compare average performances for \ee
and \ep for a set of representative values of initial beam offsets and bunch-to-
bunch jitter.

\begin{figure}[htb]
\centering
\includegraphics*[width=85mm]{MOPLS061f1.eps}
\caption{Luminosity versus vertical half beam-beam offset, for \ep and \ee
collisions simulated with GUINEA-PIG \cite{guinea}, using idealised Gaussian
beam distributions with ILC nominal parameters at 500 GeV in the center-of-
mass.}
\label{lumi-GP}
\end{figure}

\begin{figure}[htb]
\centering
\includegraphics*[width=85mm]{MOPLS061f2.eps}
\caption{Vertical deflection angle versus vertical half beam-beam offset, for
\ep and \ee collisions at the ILC with nominal parameters at 500 GeV in the
center-of-mass.}
\label{deflection-GP}
\end{figure}

\section{FEEDBACK SIMULATION}
A simplified simulation of the feedback has been carried out using parametrized
information from the last bunch crossing with a single proportional factor to
relate the measured deflection angle of the outgoing beam to the correction of
the offset at the IP \cite{feedback-Schreiber}. At frequencies of a few Hz
corresponding to the ILC train repetition rate, offsets of order of hundreds of
nm are predicted (see e.g. \cite{ground-motion}). In addition bunch-to-bunch
jitter of a fraction of the beam size can be expected. The simulation has been
done for different assumptions on the initial train offset and bunch-to-bunch
jitter, and including a 10$\%$ error on the correction to represent the measured
uncertainties. The factor relating the correction to the measured deflection
angle was optimized, independently for \ep and \ee beam parameters, to maximize
the speed of the correction without amplifying the bunch-to-bunch jitter by
over-correcting. Nominal beam parameters \cite{parameters} were used at
$\sqrt{s}=$ 500~GeV. The average luminosity loss over a train was found to be
almost independent of the offset at the beginning of the pulse for the range
considered (up to 500~nm). The \ee luminosity loss was however found to be a
factor 2 greater compared to \ep for the same assumption on the jitter.This is
due to the greater sensitivity to the vertical offset. The ability to decrease
this sensitivity with alternative beam parameters could be important if jitter
conditions are worse than expected, e.g. during early ILC operation.

With this purpose, sets of alternative beam parameters with smaller disruption
have been derived by decreasing the bunch length and varying the transverse beam
sizes, in order to maximize the luminosity while limiting the beamstrahlung
energy loss to 5$\%$ (see Table~\ref{parameters_table}). These alternative
parameters have increased luminosity, and some of them smaller sensitivity to
the IP offset, compared to those obtained for the nominal case for \ee. The
average train luminosity for different amplitudes of the jitter applied to each
beam, is also improved compared to the nominal parameters (see
Fig.\ref{feedback_eminus}).

\begin{table}[hbt]
\begin{center}
\caption{Luminosity and beamstrahlung energy loss for \ee collision for
different parameter sets with a beam energy of 250~GeV. The nominal values for
the beam sizes at the IP are $\sigma_{z0}^{*}=$300~$\mu$m and
$\sigma_{x0/y0}^{*}=$655.2/5.7~nm and the nominal intensity is
$N_{0}=2\times10^{10}$ particles.}
\vspace*{0,6cm}
\begin{tabular}{|l||c|c|c|c|c|}
\hline
\textbf{} &  \textbf{nom.} & \textbf{set 1} & \textbf{set 2} & \textbf{set 3} &
\textbf{low P} \\ \hline \hline
$N/N_{0}$                       &    1    &     1        &     1        &      1
&   0.5    \\ \hline
$\sigma_{z}^{*}$/$\sigma_{z0}^{*}$      &    1    &     0.7      &     0.5
&      0.5     &   0.5    \\
$\sigma_{x}^{*}$/$\sigma_{x0}^{*}$      &    1    &     0.7      &     0.8
&      0.9     &   0.7    \\
$\sigma_{y}^{*}$/$\sigma_{y0}^{*}$      &    1    &     1.5      &     1.5
&      1       &   0.6    \\ \hline \hline
$\epsilon_{x}^{*}$($\mu$m)              &   10    &     10       &     10
&     10       &   9.6    \\
$\epsilon_{y}^{*}$($\mu$m)              &    0.04 &      0.04    &      0.04
&      0.04    &   0.03   \\ \hline
$\beta_{x}^{*}$(mm)                     &   21.0  &     10.3     &     13.4
&     17.0     &  10.0    \\
$\beta_{y}^{*}$(mm)                     &    0.4  &      0.9     &      0.9
&      0.4     &   0.2    \\ \hline
$L~(\times10^{33})$                     &    3.9  &     4.6      &     4.9
&      5.8     &   3.0    \\
($cm^{-2}s^{-1}$)                       &         &              &
&              &          \\ \hline
$\delta_{B}$ ($\%$)                     &    2.24 &     4.9      &     5.0
&      4.3     &   2.2    \\ \hline
\end{tabular}
\label{parameters_table}
\end{center}
\end{table}

\begin{figure}[htb]
\centering
\includegraphics*[width=85mm]{MOPLS061f3.eps}
\caption{Average train luminosity normalized to the peak luminosity with nominal
parameters for \ee versus r.m.s. vertical offset difference between the beams.
The results shown include a 100~nm initial offset.}
\label{feedback_eminus}
\end{figure}

An additional set of parameters is also being investigated with only half of the
bunch charge, while keeping the same number of bunches per train. It has a
smaller peak luminosity (see Table~\ref{parameters_table}) and similar
sensitivity to IP offsets as the \ee nominal case. Such a parameter set could be
important for early ILC operation and flexibility.

\section{OPTICS STUDIES FOR THE $20\; {\rm mrad}$  CROSSING-ANGLE GEOMETRY}
\subsection{Final focus}
The optics of the Final Focus (FF) (corresponding to the 20~mrad crossing-angle
geometry) has been refitted to obtain the new $\beta$-functions at the IP for
the alternative beam parameters in Table~\ref{parameters_table}. Only the
quadrupoles upstream of the chromatic correction section and the sextupoles were
readjusted. This allows to maintain the geometry and overall optimization of
high order effects. The $\beta$-functions and the dispersion for the parameter
set 2 in Table~\ref{parameters_table} are shown in Fig.~\ref{20mrad-par2}.

\begin{figure}[h]
\hspace*{-1cm}
\rotatebox{-90}{\includegraphics*[width=65mm]{MOPLS061f4.eps}}
\caption{Optics solution for the parameter set 2, for the 20 mrad crossing-angle
geometry.}
\label{20mrad-par2}
\end{figure}

The optical bandwidth for the different sets of parameters has been studied,
considering beams with a uniform flat momentum distribution with an energy
spread of 0.1$\%$. The distribution of particles at the entrance of the FF, was
created with PLACET \cite{placet} for different average momentum offsets. The
beam was then tracked through the FF with MAD8 \cite{mad} and used as input for
GUINEA-PIG to compute the luminosity.  The results (see Fig.~\ref{bandwidth-20mrad}) show similar off-momentum behavior for all parameter sets, with the
alternative sets having better peak performance.

\begin{figure}[htb]
\centering
\includegraphics*[width=79mm]{MOPLS061f5.eps}
\caption{Optical bandwidth for the different \ee set of parameters. All the
luminosities are normalized with respect to that obtained with GUINEA-PIG at
nominal energy for nominal parameters and with ideal beam parameters at the IP
(i.e. without higher order optical effects).}
\label{bandwidth-20mrad}
\end{figure}

\subsection{Extraction line}
The effective parameters corresponding to the disrupted beam have been computed
along the extraction line for the different beam parameter sets in Table
\ref{parameters_table} (see Fig.~\ref{extraction-optics-20mrad}), and for the
different parameters sets suggested for \ep in \cite{parameters}. The largest
values found for the $\beta$-functions in the \ee and \ep cases were comparable.
The tracking of the disrupted beams has been simulated with BDSIM \cite{bdsim}
and the power losses along the line have been computed. For the parameter set 2
the losses are smaller than for the high luminosity parameters for \ep (see
Fig.~\ref{extraction-tracking-20mrad}).

\begin{figure}[htb]
\hspace{-1cm}
\rotatebox{-90}{\includegraphics*[width=65mm]{MOPLS061f6.eps}}
\caption{$\beta$-functions for the disrupted outgoing beam for the parameters
set 2.}
\label{extraction-optics-20mrad}
\end{figure}

\begin{figure}[htb]
\centering
\includegraphics*[width=84mm]{MOPLS061f7.eps}
\caption{Power losses along the extraction line for the parameter set 2 for \ee
and the high luminosity parameters, for \ep at 500 GeV in the center-of-mass.}
\label{extraction-tracking-20mrad}
\end{figure}

\section{OPTICS STUDIES FOR THE $2\; {\rm mrad}$  CROSSING-ANGLE GEOMETRY}
In the 2~mrad crossing-angle geometry the spent beam is transported off-axis
through the last defocussing quadrupole of the final focus. The kick produced by
this quadrupole is used to extract the spent beam. This scheme doesn't work for
the \ee option unless one can reverse the signs of the focusing and defocusing
final doublet quadrupoles (and sextupoles), while keeping at least the strength
of the last quadrupole to maintain the kick needed for extraction. A first
attempt in this direction \cite{snowmass-Seryi} indicated that this was
feasible, but large $\beta_{y}$-value had to be used at the IP to limit the
vertical beam size in the final doublet, which is important to keep reasonable
collimation depth. This resulted however in about a factor 2 lower peak
luminosity. Improvements with half the bunch length are also being investigated,
with for example $\beta^{*}_{x/y}=10/3$ mm. In this case, more acceptable
overall performance is expected.

 \vspace*{0,5cm}
\section{CONCLUSIONS AND PROSPECTS}
For the 20 mrad crossing-angle geometry, beam parameters can be obtained for the
\ee option by decreasing the bunch length, with improved peak luminosity,
smaller sensitivity to IP offsets, and similar beam losses in the extraction
line as those found for \ep. For the 2 mrad crossing-angle, it is necessary to
go to larger vertical beam sizes at the IP, which decreases the luminosity. With
half of the bunch length and optimizing the transverse beam sizes taking into
account collimation requirements, a first study indicates that some of this
reduction can be recovered. In the near future, these problems will be studied
to further characterize the \ee option at the ILC.

\vspace*{0,5cm}


\section{Introduction}

According to general relativity, in the asymptotic regime near spacelike singularities, a spacetime would oscillate between Kasner states.
The BKL conjectures~\cite{art:LK63,art:BKL1970,art:BKL1982} hold except where and when spikes occur~\cite{art:BergerMoncrief1993,art:GarfinkleWeaver2003}.
Spikes are a recurring inhomogeneous phenomenon in which the fabric of spacetime temporarily develops a spiky structure as the spacetime oscillates between Kasner states.
See the introduction section of~\cite{art:HeinzleUgglaLim2012} for a comprehensive background.

Previously in \cite{art:Lim2008} the orthogonally transitive (OT) $G_2$ spike solution, which is important in describing the recurring spike oscillation,
was generated by applying the Rendall-Weaver transformation~\cite{art:RendallWeaver2001} on a Kasner seed solution.
The solution is unsatisfactory, however, in that it contains permanent spikes,
and there is a debate whether permanent spike are actually unresolved spike transitions in the oscillatory regime or are really permanent.
In other words, would the yet undiscovered non-OT $G_2$ spike solution contain permanent spikes?
The proponents for permanent spikes argue that the spatial derivative terms of a permanent spike are negligible, and hence the spike stays permanent~\cite{art:Garfinkle2007}.
The opponents base their argument on numerical evidence that the permanent spike is mapped by an $R_1$ frame transition to a regime where the spatial derivative terms are not neglibigle,
which allows the spike to resolve~\cite{art:HeinzleUgglaLim2012}.
To settle the debate, we need to find the non-OT $G_2$ spike solution.
It was found that Geroch's transformation~\cite{art:Geroch1971,art:Geroch1972} would generate the desired solution, which always resolves its spike.
The next section describes the generation process.

\section{Generating the solution}

For our purpose, we express a metric $g_{ab}$ using the Iwasawa frame~\cite{art:HeinzleUgglaRohr2009}, as follows. Indicies $0,1,2,3$ corresponds to coordinates $\tau,x,y,z$.
Assume zero vorticity (zero shift). The metric components in terms of $b$'s and $n$'s are given by
\begin{align}
        g_{00} &= -N^2
\\
        g_{11} &= e^{-2b_1},\quad g_{12} = e^{-2b_1} n_1,\quad g_{13} = e^{-2b_1} n_2
\\
        g_{22} &= e^{-2b_2} + e^{-2b_1} n_1^2,\quad g_{23} =  e^{-2b_1} n_1 n_2 + e^{-2b_2} n_3
\\
        g_{33} &= e^{-2b_3} + e^{-2b_1} n_2^2 + e^{-2b_2} n_3^2.
\end{align}
One advantage of the Iwasawasa frame is that the determinant of the metric is given by
\begin{equation}
	\det g_{ab} = -N^2 e^{-2b_1-2b_2-2b_3}.
\end{equation}

A pedagogical starting point is the Kasner solution with the following parametrization:
\begin{equation}
        b_1 = \frac14(w^2-1)\tau,\quad
        b_2 = \frac12(w+1)\tau,\quad
        b_3 = -\frac12(w-1)\tau,\quad
        N^2 = {\rm e}^{-2b_1-2b_2-2b_3} = {\rm e}^{-\frac12(w^2+3)\tau},
\end{equation}
and $n_1=n_2=n_3=0$.
We shall use a linear combination of all three Killing vector fields (KVFs)
\begin{equation}
	a_1 \partial_x + a_2 \partial_y + a_3 \partial_z.
\end{equation}
as the KVF in Geroch's transformation, so that the transformation generates the most general metric possible from the given seed.

\subsection{Change of coordinates}

To simplify the KVF before applying Geroch's transformation, make the coordinate change
\begin{equation}
	x = X + n_{10} Y + n_{20} Z,\quad y = Y + n_{30} Z ,\quad z = Z
\end{equation}
where $n_{10}$, $n_{20}$, $n_{30}$ are constants.
Then the metric parameters $b_1$, $b_2$, $b_3$ and $N$ are unchanged but $n_1=n_{10}$, $n_2=n_{20}$, $n_3=n_{30}$ are now constants instead of zero.
The KVF becomes
\begin{equation}
	(a_3 (n_{10} n_{30} - n_{20}) - a_2 n_{10}  + a_1) \partial_X + (a_2 - a_3 n_{30}) \partial_Y + a_3 \partial_Z.
\end{equation}
We cannot set the $Z$ component to zero, but we can set the $X$ and $Y$ components to zero, leading to
\begin{equation}
	n_{30} = \frac{a_2}{a_3},\quad n_{10} = \frac{a_1}{a_3}.
\end{equation}
Without loss of generality, we set $a_3=1$, and so $n_{30} = a_2$ and $n_{10} = a_1$. $n_{20}$ remains free. We will see later that it can be used to eliminate any $y$-dependence.

To make transparent the effect of Geroch's transformation on the $b$'s (see (\ref{nonOT_spike})--(\ref{nonOT_spike_b3}) below), it is best to adapt the KVF to $\partial_x$.
So we make another coordinate change to swap $X$ and $Z$:
\begin{equation}
	X = \tilde{z},\quad Y = \tilde{y},\quad Z = \tilde{x},
\end{equation}
which in effect introduces frame rotations to the Kasner solution. The Kasner solution now has
\begin{align}
\label{Kasner_rotated}
	N^2 &= {\rm e}^{-\frac12(w^2+3)\tau}
\\
	{\rm e}^{-2b_1} &= {\rm e}^{(w-1)\tau} + n_{20}^2 {\rm e}^{-\frac12(w^2-1)\tau} + n_{30}^2 {\rm e}^{-(w+1)\tau}
\\
	{\rm e}^{-2b_2} &= \frac{\mathcal{A}^2}{{\rm e}^{-2b_1}}
\\
	{\rm e}^{-2b_3} &= {\rm e}^{-\frac12(w^2+3)\tau} \mathcal{A}^{-2}
\\
	n_1 &= \frac{n_{30} {\rm e}^{-(w-1)\tau} + n_{10} n_{20} {\rm e}^{-\frac12(w^2-1)\tau}}{{\rm e}^{-2b_1}}
\\
	n_2 &= \frac{n_{20} {\rm e}^{-\frac12(w^2-1)\tau}}{{\rm e}^{-2b_1}}
\\
\label{Kasner_rotated_n3}
	n_3 &= {\rm e}^{-\frac12(w^2-1)\tau}\mathcal{A}^{-2}\left[n_{30}(n_{10}n_{30}-n_{20}){\rm e}^{-(w+1)\tau}+n_{10}{\rm e}^{(w-1)\tau} \right],
\intertext{where}
\label{area}
        \mathcal{A}^2 &= (n_{10} n_{30} - n_{20})^2 {\rm e}^{-\frac12(w+1)^2\tau} + n_{10}^2 {\rm e}^{-\frac12(w-1)^2\tau} + {\rm e}^{-2\tau}.
\end{align}

Effectively, we are applying Geroch's transformation to the seed solution (\ref{Kasner_rotated})--(\ref{Kasner_rotated_n3}), using the KVF $\partial_{\tilde{x}}$.
We shall now drop the tilde from the coordinates.

\subsection{Applying Geroch's transformation}

Applying Geroch's transformation using a KVF $\xi_a$ involves the following steps.
First compute
\begin{equation}
	\lambda=\xi^a \xi_a
\end{equation}
and integrate the equation
\begin{equation}
	\nabla_a \omega =\varepsilon_{abcd}\xi ^b\nabla^c \xi^d
\end{equation}
for the general solution for $\omega$. $\omega$ is determined up to an additive constant $\omega_0$.
In our case we get
\begin{equation}
	\lambda = {\rm e}^{-2b_1} = {\rm e}^{(w-1)\tau} + {\rm e}^{-\frac12(w^2-1)\tau} n_{20}^2 + {\rm e}^{-(w+1)\tau} n_{30}^2,\quad
	\omega = 2w n_{30} z - K y + \omega_0,
\end{equation}
where the constant $K$ is given by
\begin{equation}
	K = \frac12 (w-1)(w+3) n_{20} - 2 w n_{10} n_{30}.
\end{equation}
We could absorb $\omega_0$ by a translation in the $z$ direction if $w n_{30} \neq 0$, but we shall keep $\omega_0$ for the case $w n_{30} = 0$.

The next step involves finding a particular solution for $\alpha_a$ and $\beta_a$:
\begin{align}
	\nabla_{[a}\alpha_{b]} &=\frac{1}{2}\varepsilon_{abcd} \nabla^c \xi^d,\quad \xi^a \alpha_a =\omega,
\\
	\nabla_{[a}\beta_{b]} &=2\lambda \nabla_a \xi_b + \omega \varepsilon_{abcd}  \nabla^c \xi^d,\quad \xi^a \beta_a =\omega^2 + \lambda^2 - 1.
\end{align}
Without loss of generality, we choose $\theta=\frac{\pi}{2}$ in Geroch's transformation, so $\alpha_a$ is not needed in $\eta_a$ below.
We assume that $\beta_a$ has zero $\tau$-component. Its other components are
\begin{align}
	\beta_1 &= \omega^2 + \lambda^2 -1
\\
        \beta_2 &= n_{10} n_{20}^3 {\rm e}^{-(w^2-1)\tau} + \left[ 2 \frac{w-1}{w+1} n_{10} n_{20} n_{30}^2 + \frac{4}{w+1} n_{20}^2 n_{30} \right] {\rm e}^{-\frac12(w+1)^2 \tau}
\notag\\
        &\quad
                + 2 \frac{w+1}{w-1} n_{10} n_{20} {\rm e}^{-\frac12(w-1)^2\tau} + (w+1) n_{30} {\rm e}^{-2\tau} + n_{30}^3 {\rm e}^{-2(w+1)\tau} + F_2(y,z)
\\
        \beta_3 &= n_{20}^3 {\rm e}^{-(w^2-1)\tau} + 2 n_{20} n_{30}^2 \frac{w-1}{w+1} {\rm e}^{-\frac12(w+1)^2 \tau} + 2 n_{20} \frac{w+1}{w-1} {\rm e}^{-\frac12(w-1)^2\tau} + F_3(y,z)
\end{align}
where $F_2(y,z)$ and $F_3(y,z)$ satisfy the constraint equation
\begin{equation}
        - \partial_z F_2 + \partial_y F_3 + 2(w-1)\omega = 0.
\end{equation}
For our purpose, we want $F_3$ to be as simple as possible, so we choose
\begin{equation}
	F_3 = 0,\quad F_2 = \int 2(w-1)\omega {\rm d} z = 2w(w-1) n_{30} z^2 - 2 (w-1) K y z +2(w-1)\omega_0 z.
\end{equation}

The last step constructs the new metric. Define $\tilde{\lambda}$ and $\eta_a$ as
\begin{align}
	\frac{\lambda}{\tilde{\lambda}} &= (\cos\theta-\omega\sin\theta)^2 +\lambda^2 \sin^2\theta,
\\
	\eta_a &=\tilde{\lambda}^{-1} \xi_a +2 \alpha_a \cos\theta\sin\theta-\beta_a \sin^2\theta.
\end{align}
The new metric is given by
\begin{equation}
	\tilde{g}_{ab}=\frac{\lambda}{\tilde{\lambda}}(g_{ab}-\lambda^{-1} \xi_a\xi_b)+\tilde{\lambda} \eta_a \eta_b.
\end{equation}
In our case $\tilde{g}_{ab}$ is given by the metric parameters
\begin{align}
\label{nonOT_spike}
        \tilde{N}^2 &= N^2 (\omega^2+\lambda^2)
\\
        {\rm e}^{-2\tilde{b}_1} &= \frac{{\rm e}^{-2b_1}}{\omega^2+\lambda^2}
\\
        {\rm e}^{-2\tilde{b}_2} &= {\rm e}^{-2b_2} (\omega^2+\lambda^2)
\\
\label{nonOT_spike_b3}
        {\rm e}^{-2\tilde{b}_3} &= {\rm e}^{-2b_3} (\omega^2+\lambda^2)
\\
        \tilde{n}_1 &= -2w(w-1) n_{30} z^2 + 2 (w-1) K y z - 2(w-1)\omega_0 z + \frac{\omega^2}{\lambda}(n_{30} {\rm e}^{-(w+1)\tau} + n_{10} n_{20} {\rm e}^{-\frac12(w^2-1)\tau})
\notag\\
        &\quad  -\left[ n_{30} w {\rm e}^{-2\tau} + \frac{w+3}{w-1} n_{10} n_{20} {\rm e}^{-\frac12(w-1)^2\tau} + \frac{w-3}{w+1} n_{20} n_{30} (n_{10} n_{30} - n_{20}) {\rm e}^{-\frac12(w+1)^2\tau} \right]
\\
        \tilde{n}_2 &= n_{20} {\rm e}^{-\frac12(w^2-1)\tau} \left[ - \frac{w+3}{w-1} {\rm e}^{(w-1)\tau} - n_{30}^2 \frac{w-3}{w+1} {\rm e}^{-(w+1)\tau} + \frac{\omega^2}{\lambda} \right]
\\
        \tilde{n}_3 &= \mathcal{A}^{-2} \left[ n_{10} {\rm e}^{-\frac12(w-1)^2\tau} + n_{30} (n_{10} n_{30} - n_{20}) {\rm e}^{-\frac12(w+1)^2\tau} \right],
\label{nonOT_spike_n3}
\end{align}
and $\mathcal{A}$, given by (\ref{area}), is the area density~\cite{art:vEUW2002} of the $G_2$ orbits.
Note that the $w=\pm1$ cases would have to be computed separately, which we shall leave to future work.
The new solution admits two commuting KVFs:
\begin{equation}
        \partial_x,\quad [- (w-1)K^2y^2+2(w-1)K\omega_0 y] \partial_x + 2wn_{30}\partial_y + K \partial_z.
\end{equation}
Their $G_2$ action is non-OT, unless $n_{10}=n_{20}=0$.
The solution is also the first non-OT Abelian $G_2$ explicit solution found.

In the next section we shall focus on the case where $K=0$, or equivalently, where
\begin{equation}
\label{n20choice}
	n_{20} = \frac{4w}{(w-1)(w+3)} n_{10} n_{30},
\end{equation}
which turns off the $R_2$ frame transition (which is shown  to be asymptotically suppressed in~\cite{art:HeinzleUgglaRohr2009}), and eliminates the $y$-dependence.
Setting (\ref{n20choice}) in the rotated Kasner solution (\ref{Kasner_rotated})--(\ref{Kasner_rotated_n3}) also turns off the $R_2$ frame transition there, giving
the explicit solution that describes the double frame transition $\mathcal{T}_{R_3 R_1}$ in~\cite{art:HeinzleUgglaRohr2009}.
The mixed frame/curvature transition $\mathcal{T}_{N_1 R_1}$ in~\cite{art:HeinzleUgglaRohr2009} is described by the metric $\tilde{g}_{ab}$
with $n_{20} = n_{30} = 0$. Both the double frame transition and the mixed frame/curvature transition are encountered in the exceptional Bianchi type VI$_{-1/9}^*$
cosmologies~\cite{art:Hewittetal2003}.

Setting $n_{10}=n_{20}=0$ yields the OT $G_2$ spike solution in~\cite{art:Lim2008}. To adapt the solutions in~\cite{art:Lim2008} to the Iwasawa frame here, let
\begin{equation}
	b_1 = - \frac12 (P(\tau,z)-\tau), \quad b_2 = \frac12 (P(\tau,z)+\tau), \quad b_3 = - \frac14 (\lambda(\tau,z)+\tau), \quad n_1 = -Q(\tau,z), \quad n_2=n_3=0,
\end{equation}
where $x$-dependence in~\cite{art:Lim2008} becomes $z$-dependence here, and set $w$ to $-w$, $\lambda_2=\ln 16$, $Q_0 = 1$, $Q_2 = 0$ there, and set $n_{30}=1$, $\omega_0=0$ here.
As pointed out in~\cite{art:Limetal2009} and~\cite{art:Binietal2009}, the factor 4 in Equation (34) of~\cite{art:Lim2008} should not be there.

\section{The dynamics of the solution}

To describe the dynamics of the non-OT spike solution, we shall plot the state space orbit
projected onto the Hubble-normalized $(\Sigma_+,\Sigma_-)$ plane, as done in~\cite{art:Lim2008}.
The formulas are
\begin{align}
	\Sigma_+ &= -1 + \frac14 \mathcal{N}^{-1} \partial_\tau (\mathcal{A}^2)
\\
	\Sigma_- &= \frac{1}{2\sqrt{3}} \mathcal{N}^{-1} \partial_\tau(\tilde{b}_2 - \tilde{b}_1)
\\
	\mathcal{N} &= \frac16\left[ \frac{\partial_\tau (\lambda^2)}{\omega^2+\lambda^2} + \partial_\tau \ln (N^2) \right]
\end{align}
\cite{art:HeinzleUgglaRohr2009} uses a different orientation, where their $(\Sigma_+,\Sigma_-)$ are given by
\begin{align}
	\Sigma_+ &= -\frac12(\Sigma_+ + \sqrt{3}\Sigma_-)
\\
	\Sigma_- &= -\frac12(\sqrt{3}\Sigma_+ - \Sigma_-)
\end{align}

The non-OT spike solution (with $K=0$, $\omega_0=0$) goes from a Kasner state with $2 < w < 3$, through a few intermediate Kasner states, and arrives at the final Kasner state with $w < -1$.
The transitions are composed of spike transitions and $R_1$ frame transitions.
The non-OT spike solution always resolves its spike, unlike the OT spike solution with $|w|<1$, which has a permanent spike.

For a typical Kasner source with $2 < w < 3$, there are six non-OT spike solutions, some of which are equivalent, that start there. For example,
non-OT spike solutions with $|w| = \tfrac13,\ 2,\ 5$ all start at $w_\text{source} = \frac73$. From there, however, there are two extreme alternative spike orbits. The first
alternative is to form a ``permanent" spike, followed by an $R_1$ transition, and lastly to resolve the spike. This was described in~\cite{art:HeinzleUgglaLim2012} as the joint spike transition.
This alternative is more commonly encountered (assuming that permanent spikes are more commonly encountered than no-spike at the end of a Kasner era).
The second alternative is to undergo an $R_1$ transition first, followed by a transient spike transition, and finish with another $R_1$ transition.
By varying $n_1$ and $n_3$, one can get orbits that are close to one extreme alternative or the other, or some indistinct mix.

The sequence of $w$-value of the Kasner states for the spike orbit is given below.
For non-OT spike solution with $|w|>3$, the first and second alternatives are
\begin{align}
        &\frac{3|w|-1}{1+|w|},\ \frac{5+|w|}{1+|w|},\ 2+|w|,\ 2-|w|
\\
        &\frac{3|w|-1}{1+|w|},\ \frac{3|w|+1}{|w|-1},\ \frac{|w|-5}{|w|-1},\ 2-|w|
\end{align}
For $1<|w|<3$, the first and second alternatives are
\begin{align}
        &\frac{5+|w|}{1+|w|},\ \frac{3|w|-1}{1+|w|},\ \frac{3|w|+1}{|w|-1}, \frac{5-|w|}{1-|w|}
\\
        &\frac{5+|w|}{1+|w|},\ 2+|w|,\ 2-|w|, \frac{5-|w|}{1-|w|}
\end{align}
For $|w|<1$, the first and second alternatives are
\begin{align}
        &\frac{5+|w|}{1+|w|},\ \frac{3|w|-1}{1+|w|},\ \frac{3|w|+1}{|w|-1}, \frac{5-|w|}{1-|w|}
\\
        &\frac{5+|w|}{1+|w|},\ 2+|w|,\ 2-|w|, \frac{5-|w|}{1-|w|}
\end{align}
For $|w|<1$, the first and second alternatives are
\begin{align}
        &2+|w|,\ 2-|w|,\ \frac{5-|w|}{1-|w|},\ \frac{3|w|+1}{|w|-1}
\\
        &2+|w|,\ \frac{5+|w|}{1+|w|},\ \frac{3|w|-1}{1+|w|},\ \frac{3|w|+1}{|w|-1}
\end{align}
For example, for $|w| = \tfrac13,\ 2,\ 5$, the first alternative is $\frac73,\ \frac53,\ 7,\ -3$ and the second alternative is $\frac73,\ 4,\ 0,\ -3$.
See Figure~\ref{fig:w5alternatives}.

\begin{figure}
  \begin{center}
    \resizebox{\textwidth}{!}{\includegraphics{w5alternatives_v2_cropped.png}}
    \caption{Alternative spike orbits for $w=5$. Top row is the orientation used in~\cite{art:Lim2008}, bottom row is the orientation used in~\cite{art:HeinzleUgglaRohr2009}.
Left column is the first alternative orbit, right column is the second alternative.
Spike orbits ($z=0$) are in red, faraway orbits ($z=10^{12}$) in blue.
Left column is generated with $n_{10}=10^{-3}$, $n_{30}=1$, right column with $n_{10}=10^9$, $n_{30}=10^{-9}$.
A red circle marks the start of the orbits, a red star marks the end.}
    \label{fig:w5alternatives}
\end{center}
\end{figure}

\section{Summary}

In this paper, we went through the steps of generating the non-OT $G_2$ spike solution, and illustrated its state space orbits for the case $K=0$, which show two extreme alternative orbits.
More importantly, the non-OT $G_2$ spike solution always resolves its spikes, in contrast to its OT $G_2$ special case which produces an unresolved permanent spike for some parameter values.
The non-OT $G_2$ spike solution shows that, in the oscillatory regime near spacelike singularities, unresolved permanent spikes are artefacts of restricting oneself to the OT $G_2$ case,
and that spikes are resolved in the more general non-OT $G_2$ case. Therefore spikes are expected to recur in the oscillatory regime rather than to become permanent spikes.
We also obtained explicit solutions describing the double frame transition and the mixed frame/curvature transition in~\cite{art:HeinzleUgglaRohr2009}.
We leave the further analysis of the non-OT $G_2$ spike solution to future work.

\section*{Acknowledgment}

Part of this work was carried out at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) and Dalhousie University.
I would like to thank Claes Uggla and Alan Coley for useful discussions.
The symbolic computation software {\tt MAPLE} and numerical software {\tt MATLAB} are essential to the work.