\section{Introduction}
Jet clustering algorithms, which map the particles observed in the final state
of a high-energy collisions into a smaller number of (usually) well defined
objects -- the jets -- are widely used in the study of the properties of strong
interactions. The jets are usually meant to be good proxies of the original
partons (though the detailed relation is more subtle), and by studying them one
tries to probe the underlying dynamics. The reason for using the jets, rather
than directly the observed hadrons, is that they can be construed as
infrared-safe observables: they are therefore amenable to perturbative QCD
predictions, and their sensitivity to non-perturbative phenomena
(hadronisation, underlying event and pileup effects) can either be kept
under control or corrected for.

In this talk we explore the issue of the susceptibility of jets to
contamination from soft radiation distributed in the form of a roughly uniform
and diffuse background. Physical examples are the pileup originated by multiple
minimum bias collisions in high-luminosity hadron colliders like the LHC, the
many particles produced in a central heavy ion collision and, to a lesser
extent, the underlying event given by perturbative and non-perturbative
QCD radiation whenever strongly-interacting particles are produced at high
energy. We shall argue that this susceptibility can be quantitatively
characterised in terms of the novel concept of {\sl area of a jet}, which we
shall rigorously introduce. In turn, this will suggest a procedure by means
of which such contamination can be subtracted from the jet momentum, so as to
recover -- to a large extent -- its proxy relation with the parton it
originated from.

Naively, one can think of the jet area as the surface (in the rapidity-azimuth
plane) over which the particles that have been clustered into a given jet are
distributed. One can also assume that the amount of diffuse
background radiation clustered together with the jet will be
proportional to this area. One could therefore think of determining somehow the
momentum surface density of this noise, $\rho$, and successively subtract
from the jet momentum a quantity given by $\rho$ times the area of the jet.

Before such a program can be implemented in practice, however, the jet area
needs to be defined more rigorously, and a procedure to extract $\rho$ must be
devised. This is done in \cite{css} and \cite{cs} respectively, where
both aspects are introduced and extensively studied.

\section{Jet Area}
The naive vision of the jet area as the surface covered by the particles that
make up the jet quickly turns out to be fallacious: as the particles are
point-like, this area is zero. Drawing some sort of boundary, like for instance
a convex hull -- the minimal set of particles such that all the others are
contained in the polygon drawn through them  -- is also prone to ambiguities:
different jets may overlap, and a region of space might be arbitrarily assigned
to a jet irrespectively of the properties of the clustering algorithm.

\begin{figure}
  \centering
  \includegraphics[height=5cm, width=0.48\textwidth]{areas-hist.eps}~~~
  \includegraphics[height=5cm, width=0.48\textwidth]{soft-hard.eps}
  \vspace{-.2cm}
  \caption{a) Active area distributions for the $k_t$
  algorithm~\protect\cite{kt}. Cambridge/Aachen~\protect\cite{cam} has a
  very similar behaviour. b) Average are of jet containing a hard
  particle as a function of the ratio of its momentum to that of the soft
  background jets.}
  \label{fig:areas}
 \vspace{-.4cm}
 \end{figure}

To overcome these difficulties, we propose a definition of jet area which is
inherently related to the clustering procedure, and which can properly account
for the jet contamination due to a diffuse background. Our definition is
strictly dependent on the infrared-safety property that a good jet algorithm
should have: the addition of one (or many) soft particles to the event should
not change the final set of hard jets. We add therefore a large number of uniformly
distributed and extremely soft particles ({\sl ghosts}) to the event,
and cluster them together with the real particles. At the end of the clustering
procedure, the number of ghosts clustered with each jet will provide a robust
measure of the jet's extension in the rapidity-azimuth plane, and define
therefore its {\sl active area}, $A$.\footnote{The drawback of this procedure is that a very
large number of particles needs to be clustered (a few thousands ghosts are
needed to achieve accuracies of the order of one per cent). This would be
unfeasible -- or at least extremely unpractical -- without the fast implementations
of the $k_t$~\cite{kt} and the Cambridge/Aachen~\cite{cam}
jet algorithms provided by {\tt FastJet}~\cite{Cacciari:2005hq}.
This package also provides the tools to calculate the area of the jets, as
well as an interface to the new infrared-safe cone algorithm
SISCone~\cite{siscone}.}

Fig.~\ref{fig:areas}(a) shows how the values for this active area are
distributed for two kinds of events: on one extreme, jets constituted
of many uniformly distributed particles with similar momenta (the pure-ghost jets);
on the other extreme, a jet containing a single hard particle. We can
see that these two situations produce different distributions
for the active areas, with different averages: the jets
containing many similar particles have a typical area of
order $\langle A^{soft}\rangle \simeq 0.55\,\pi R^2$ ($R$ is the typical
radius parameter present in most jet algorithms), while the jets
containing a single hard particle tend to be larger, their average area
being  $\langle A^{hard}\rangle \simeq 0.81\, \pi R^2$.

One can take farther this exploration of similarities and differences between
soft (i.e. uniform) and hard jets, and explore how the transition takes place:
fig.~\ref{fig:areas}(b) shows the average area of the jet containing the single
``hard'' particle as its transverse momentum $p_t$ changes from being
negligible with respect to the soft background to being much larger. One can
see that in the $p_{t,\,hard} \gg \langle p_{t,\,soft~jet}\rangle$ limit the
$\simeq 0.81\, \pi R^2$ value for the average area is recovered. On the other
hand, in the opposite  $p_{t,\,hard} \ll \langle p_{t,\,soft~jet}\rangle$ limit
the ``hard'' jet now behaves like a soft one, the difference in average area
being only of probabilistic nature related to the ``measurement'' of the area
of the specific jet containing a given particle.

\begin{figure}[pt]
  \centering
  \includegraphics[height=5cm, width=0.48\textwidth]{scatter-pt-v-area.eps}~~
  \includegraphics[height=5cm, width=0.48\textwidth]{scatter.eps}
  \vspace{-.2cm}
  \caption{A dijet event superimposed to 10 minimum bias events originated by
  moderate-luminosity pileup in $pp$ collisions at the LHC, as simulated
  by PYTHIA.}
  \label{fig:rho}
  \vspace{-.4cm}
\end{figure}

\section{Noise Level}
\label{sec:rho}
The estimation of $\rho$, the typical level of the background radiation, could
probably be performed in many ways. The method we propose here is related to
the jet areas discussed above. It relies on the observation that the transverse
momentum of a jet divided by its area, $p_{ti}/A_i$, behaves differently for the
hard jets and for the background ones. Typically, the jets originating from the
background radiation cluster themselves in a band, while the hard jets
stick out. This is clearly shown in fig.~\ref{fig:rho}. This event is a
simulated $pp$ collision at the LHC at moderate luminosity: 10 additional minimum
bias events are added to the main hard collision, which produces a dijet event
with jets of transverse momentum of the order of 50 GeV. Fig.~\ref{fig:rho}
(left) shows that the areas of the various jets can fluctuate widely. However,
when the same jets are plotted in terms of $p_{ti}/A_i$ (right plot)
one clearly see the
band established by the background. Different strategies can be devised to
quantitatively determine its level. One of the simplest one is to take the
{\sl median} of all the $p_{ti}/A_i$, an operation that prevents
the few hard jets from biasing its value. We define therefore:
\begin{equation}
\label{eq:rho}
\rho = \mathrm{median}\left[\left\{\frac{p_{ti}}{A_i}\right\}\right] \; .
\end{equation}
In the specific case of the event of fig.~\ref{fig:rho}, the momentum density of the
background is therefore $\rho \simeq 6$~GeV per unit area.

\section{Background Subtraction}
Once the area of each jet, $A_i$, and the noise level $\rho$ are known, one can
correct the transverse momentum via the
following operation:
\begin{equation}
\label{eq:subtraction}
p_{ti}^{\mathrm{(sub)}} = p_{ti} - \rho A_i \; .
\end{equation}
We show how this works in practice by considering the following toy model: we
generate many events which contain a single hard particle, with a transverse
momentum $p_t^{hard} = 100$~GeV,  embedded in a background of
10000 soft particles, each with an average transverse momentum $\langle
p_t^{soft}\rangle = 1$~GeV (with little fluctuations, 10\%, around this
value) and
randomly uniformly distributed in rapidity and azimuth up to $y_{max} = 4$. In
this particular case we can of course calculate
the transverse momentum density (per unit area) of the soft  particles
from the input parameters, since we know how we generated them:
\begin{equation}
\label{eq:rho-toy}
\rho = \Big\langle \frac{dp_t^{soft}}{dy\,d\phi} \Big\rangle = \frac {10000
\times 1\,\mathrm{GeV} }{2 \times y_{max} \times 2\pi} \simeq
200~\mathrm{GeV} \, .
\end{equation}
This situation might look extreme, but similar values are expected in realistic
cases, like a central Pb Pb collision at the LHC.

\begin{floatingfigure}{0.48\textwidth}
  \centering
  \includegraphics[height=5cm, width=0.48\textwidth]{soft-hard-subtraction.eps}
  \caption{Jets containing a hard particle with $p_{t,\,hard} = 100$~GeV
  clustered together with a soft background (green, ``raw'' histogram),
  and after its subtraction (blue ``corrected'' one). The `4-vector'
  versions of the area and of the subtraction~\protect\cite{css,cs}, more
  appropriate for large $R$, have been used
  for this plot.}
  \label{fig:subtraction}
  \vspace{.1cm}
\end{floatingfigure}

We know from the previous section that an average soft jet, when
clustered with the $k_t$ or the Cambridge/Aachen algorithm with $R=1$,
has an area of order $0.55 \pi$. This translates in a typical
transverse momentum $p_t^{soft~jet} \simeq \rho \langle A^{soft}
\rangle \simeq 350~\mathrm{GeV}$. Such jets would already dwarf the
hard particle of 100 GeV. However, this particle will itself be
embedded in a jet containing also many soft particles: this jet will
therefore have a typical transverse momentum of the order of 350 + 100
GeV, but huge fluctuations will be visible from one event to another,
as the amount of background clustered with it will vary considerably.

This means that both the absolute energy scale and the energy
resolution are degraded by the presence of the background, as shown in
fig.~\ref{fig:subtraction}: the transverse momentum of the hard jet is
displaced, by an amount consistent with our estimate,  and the
resolution is hopelessly bad (green histogram, ``raw'').   However,
once the subtraction is performed according to
eq.~(\ref{eq:subtraction}) (using for each event the $\rho$ directly
extracted from the clustering, as explained in Sec.~\ref{sec:rho}, and
{\sl not} the fixed value of eq.~(\ref{eq:rho-toy}), of course), the
correct average transverse momentum is recovered, together with a large
fraction of the resolution (blue histogram, ``corrected'').

This toy model shows the feasibility and the accuracy of the
determination of the noise level and of the subtraction
procedure. More realistic examples, and references to experimental
investigations of the problem of background subtraction, are given in \cite{cs}.

{\bf Acknowledgements.} This work has been performed, and is partially in
progress, with Gavin Salam and Gregory Soyez, whom I thank for an extremely
stimulating collaboration.
\vspace{-.5cm}
\section*{References}


\section*{Acknowledgments}
We thank H. Chen, A. E. Hosoi, C. Pipe, M. Roper and H. Stone for useful discussions. This work was supported in part by the Charles Reed Fund at MIT and by the National Science Foundation (CTS-0624830)


\section{Introduction}

In recent papers the tilted Bianchi models
\cite{HBWII,HWV,Harnett,BHtilted,hervik,CH1,CH2,HHC,HHLC,HLim,HHLC2} have been studied using
dynamical systems methods \cite{DS1,DS2}.  In this paper we will study the
general tilted\footnote{The non-tilted Bianchi model of type VI$_{-1/9}$ was studied in \cite{HHW-19}.} perfect fluid
Bianchi model of type VI$_{-1/9}$, with a constant linear equation of state parameter
$\gamma$ satisfying the causality conditions (i.e., no superluminal speed of sound),
which was not studied in detail in \cite{HHLC}.  The
tilted models have shown a wide variety of phenomena including, for example, the existence
of closed curves
\cite{HHC} and incompleteness of the fluid congruence \cite{CollinsEllis,tilt1}.

The general irrotational perfect fluid type VI$_{-1/9}$ model is the most general  of
the irrotational Bianchi models.  In fact, the irrotational type VI$_{-1/9}$ model has
a 6-dimensional state space; just one dimension less than that of of the most general tilted type
VI$_{-1/9}$ model.  We will show that, regardless of whether we consider the general type
VI$_{-1/9}$ model with or without vorticity, the important late time asymptotes are the
non-tilted Collins type VI$_{-1/9}$ solution (for $2/3<\gamma<10/9$), \[ \d s^2=-\d
t^2+t^2\d x^2+t^{\frac{2(4-3\gamma)}{3\gamma}}e^{-\frac{2sx}{3\gamma}}\d
y^2+t^{\frac{4}{3\gamma}}e^{\frac{4sx}{3\gamma}}\d z^2, \] where
$s=\sqrt{(2-\gamma)(3\gamma-2)}$, or the Collinson-French (or Robinson-Trautman) vacuum
solution (for $10/9<\gamma<2$), \[ \d s^2=-\d t^2+t^2\d x^2+t^{\frac
25}\left(e^{-\frac{\sqrt{6}}{5}x}\d y+\frac{\sqrt{5}}2t^{\frac 45}\d
x\right)^2+t^{\frac 65}e^{\frac{4\sqrt{6}}{5}x}\d z^2.\] The tilt of the fluid can be
asymptotically non-tilted, intermediate or extreme depending on the value of $\gamma$.
The Collinson-French solution is a Petrov type III vacuum solution, and hence it
has the peculiar feature that all the curvature invariants of zeroth order
vanish\footnote{However, considering differential invariants they will not vanish; for
example, $C_{\alpha\beta\gamma\delta;\mu}C^{\alpha\beta\gamma\delta;\mu}\neq 0$.}.

The spatial hypersurfaces in a Bianchi type VI$_{-1/9}$ cosmology are defined as the
orbit of the action of the Bianchi type VI$_{-1/9}$ Lie group.  This Lie group can be
equipped with the left-invariant one-forms:  \[ \widetilde{\mbold\omega}^1=\d x, \quad
\widetilde{\mbold\omega}^2=e^{-bx}\d y,\quad \widetilde{\mbold\omega}^3=e^{2bx}\d z,\]
where $b$ is a constant.  The exceptional feature of this model does not relate to the
group itself (for which there is nothing extraordinary) but arises from the Einstein field
equations.  For this particular model one of the Einstein constraint equations vanishes
exactly and hence the vacuum case allows for an extra shear degree of freedom.  This
extra shear is present in the Collinson-French vacuum solution and therefore this metric
has no
analogue in the other Bianchi vacuum cosmologies.

The aim of this paper is to fill one of the gaps in the analysis of the general
tilted type VI$_h$ models presented in
\cite{HHLC}.  The $h=-1/9$ case is special (as explained above) and requires a separate
analysis, which has only been done in part previously.  In particular,  we will show
that closed periodic orbits exist for the general vortic model.  However, for the
irrotational models, no such closed curve exists.  Moreover, we will also
complete a centre manifold analysis appearing in one 6-dimensional invariant subspace
describing vortic Bianchi type VI$_{-1/9}$ models.  All other attractors will be
included for completeness.

\section{Equations of motion}
\subsection{The orthonormal frame approach}
The line-element of a Bianchi cosmology can be written
\begin{eqnarray}
\d s^2=-\d t^2+\delta_{ab}{\mbold\omega}^a{\mbold\omega}^b,
\end{eqnarray}
where $t$ is the co-moving cosmological time. The one-forms ${\mbold\omega}^a$ are left-invariant one-forms on the hypersurfaces spanned by the group orbits and can be related to the left-invariant one-forms, $\widetilde{\mbold\omega}^i$, given above by ${\mbold\omega}^a={\sf e}^a_{~i}(t)\widetilde{\mbold\omega}^i$.

The geometric (or normal) congruence, $n^{\mu}$, is given by ${\bf n}=\partial/\partial t$. It is also useful to define and the Hubble scalar and the shear associated with the congruence $n^\mu$:
\begin{eqnarray}
H\equiv\frac 13n^\mu_{~;\mu}, \quad \sigma_{\mu\nu}\equiv n_{\mu;\nu}-Hh_{\mu\nu}.
\end{eqnarray}
The matter variables are chosen to be the energy density, $\mu$, and the tilt-velocity, $v^a$, which is defined as the 3-velocity of the fluid with respect to the geometric (or normal) congruence, $n^{\mu}$.
The equations of motion can now be written down in terms of the Hubble scalar, $H;$ the shear, $\sigma_{ab}$; the curvature variables $n^{ab}$ and $a_c$; and the matter variables $\mu$ and $v^a$.

Expansion-normalised variables are introduced  (by dividing the variables with the appropriate powers of $H$).
The papers \cite{CH2,HHLC2} contain all the details regarding the  determination of the evolution
equations for the tilted cosmological models under consideration.
Here, we shall adopt the so-called $N$-gauge in which the function
${\bf N}_{\times}$ is purely imaginary; this is realised by the choice
$\phi'=\sqrt{3}\lambda\Sigma_-$, where $\lambda$ is defined by
$\bar{N}=\lambda\mathrm{Im}({\bf N}_{\times})$. The evolution equation for
$\bar{N}$ can then be replaced by an evolution equation for $\lambda$,
which ensures a closed system of equations.
We will also adopt the dimensionless time parameter $\tau$, which is related to the cosmological time $t$ via $\mathrm{d} t/\mathrm{d}\tau =(1/H)$, where $H$ is the Hubble scalar.

Using expansion-normalised variables, the equations of motion
are (see \cite{CH2,HHLC2} for the complete derivation of the equations):
\begin{eqnarray}
\Sigma_+'&=& (q-2)\Sigma_++{3}(\Sigma_{12}^2+\Sigma^2_{13})-2N^2 +\frac{\gamma\Omega}{2G_+}\left(-2v_1^2+v_2^2+v_3^2\right) \\
\Sigma_-'&=&(q-2-2\sqrt{3}\Sigma_{23}\lambda)\Sigma_-+\sqrt{3}(\Sigma_{12}^2-\Sigma_{13}^2) +2AN +\frac{\sqrt{3}\gamma\Omega}{2G_+}\left(v_2^2-v_3^2\right)\\
\Sigma'_{12}&=& \left(q-2-3\Sigma_+-\sqrt{3}\Sigma_-\right)\Sigma_{12} -\sqrt{3}\left(\Sigma_{23}+\Sigma_-\lambda\right)\Sigma_{13} +\frac{\sqrt{3}\gamma\Omega}{G_+}v_1v_2\\
\Sigma'_{13}&=&\left(q-2-3\Sigma_++\sqrt{3}\Sigma_-\right)\Sigma_{13}-\sqrt{3}\left(\Sigma_{23}-\Sigma_-\lambda\right)\Sigma_{12}+\frac{\sqrt{3}\gamma\Omega}{G_+}v_1v_3\\
\Sigma'_{23}&=&(q-2)\Sigma_{23}-2\sqrt{3}N^2\lambda+2\sqrt{3}\lambda\Sigma_-^2+2\sqrt{3}\Sigma_{12}\Sigma_{13}+ \frac{\sqrt{3}\gamma\Omega}{G_+}v_2v_3\\
N'&=& \left(q+2\Sigma_++2\sqrt{3}\Sigma_{23}\lambda\right){N}\\
\lambda' &=& 2\sqrt{3}\Sigma_{23}\left(1-\lambda^2\right)\\
A'&=& (q+2\Sigma_+)A .
\end{eqnarray}
The equations for the fluid are:
\begin{eqnarray}
\Omega'&=& \frac{\Omega}{G_+}\Big\{2q-(3\gamma-2)+2\gamma Av_1
 +\left[2q(\gamma-1)-(2-\gamma)-\gamma\mathcal{S}\right]V^2\Big\}
 \quad \\
 v_1' &=& \left(T+2\Sigma_+\right)v_1-2\sqrt{3}\Sigma_{13}v_3-2\sqrt{3}\Sigma_{12}v_2-A\left(v_2^2+v_3^2\right)-\sqrt{3}N\left(v_2^2-v_3^2\right)\\
 v_2'&=& \left(T-\Sigma_+-\sqrt{3}\Sigma_-\right)v_2-\sqrt{3}\left(\Sigma_{23}+\Sigma_-\lambda\right)v_3+\sqrt{3}\lambda{N}v_1v_3+\left(A+\sqrt{3}N\right)v_1v_2 \\
 v_3'&=& \left(T-\Sigma_++\sqrt{3}\Sigma_-\right)v_3-\sqrt{3}\left(\Sigma_{23}-\Sigma_-\lambda\right)v_2-\sqrt{3}\lambda{N}v_1v_2+\left(A-\sqrt{3}N\right)v_1v_3 \\
 V'&=&\frac{V(1-V^2)}{1-(\gamma-1)V^2}\left[(3\gamma-4)-2(\gamma-1)Av_1-\mathcal{S}\right],
\end{eqnarray}
where
\begin{eqnarray} q&=& 2\Sigma^2+\frac
12\frac{(3\gamma-2)+(2-\gamma)V^2}{1+(\gamma-1)V^2}\Omega\nonumber \\
\Sigma^2 &=& \Sigma_+^2+\Sigma_-^2+\Sigma_{12}^2+ \Sigma_{13}^2+\Sigma_{23}^2\nonumber \\
\mathcal{S} &=& \Sigma_{ab}c^ac^b, \quad c^ac_{a}=1, \quad v^a=Vc^a,\quad \nonumber \\
 V^2 &=& v_1^2+v_2^2+v_3^2,\quad  \nonumber \\
 T&=& \frac{\left[(3\gamma-4)-2(\gamma-1)Av_1\right](1-V^2)+(2-\gamma)V^2\mathcal{S}}{1-(\gamma-1)V^2}\nonumber\\
 G_+&=&1+(\gamma-1)V^2\nonumber.
\end{eqnarray}
These variables are subject to the constraints
\begin{eqnarray}
1&=& \Sigma^2+A^2+N^2+\Omega \label{const:H}\\
0 &=& 2\Sigma_+A+2\Sigma_-N+\frac{\gamma\Omega v_1}{G_+} \label{const:v1}\\
0 &=&
-\left[\Sigma_{12}(N+\sqrt{3}A)+\Sigma_{13}\lambda{N}\right]+\frac{\gamma\Omega v_2}{G_+} \label{const:v2}\\
0 &=&
\left[\Sigma_{13}(N-\sqrt{3}A)+\Sigma_{12}\lambda{N}\right]+\frac{\gamma\Omega v_3}{G_+} \label{const:v3} \\
0&=& 3A^2-\left(1-\lambda^2\right)N^2.\label{const:group}
\end{eqnarray}
The parameter $\gamma$ will be assumed to satisfy $\gamma\in (0,2)$.
The generalized Friedmann equation (\ref{const:H}) yields an expression which effectively defines the energy density $\Omega$. We will assume that this energy density is non-negative: $\Omega\geq 0$.
Therefore, the state vector can be considered
${\sf X}=[\Sigma_+,\Sigma_-,\Sigma_{12},\Sigma_{13},\Sigma_{23},N,\lambda,A,v_1,v_2,v_3]$
modulo the constraint equations  (\ref{const:v1})-(\ref{const:group}).
Thus the dimension of the physical state space is seven
(and hence of equal generality to the other models of type VI$_h$ for a given value of $h$).
Additional details are presented in \cite{CH2}.

The dynamical system is invariant under the following discrete symmetries :
$$\begin{tabular}{l}
$\phi_1:~[\Sigma_+,\Sigma_-,\Sigma_{12},\Sigma_{13},\Sigma_{23},N,\lambda,A,v_1,v_2,v_3]
\mapsto  [\Sigma_+,\Sigma_-,\Sigma_{12},\Sigma_{13},\Sigma_{23},-N,\lambda,-A,-v_1,-v_2,-v_3] $ \\
$\phi_2:~[\Sigma_+,\Sigma_-,\Sigma_{12},\Sigma_{13},\Sigma_{23},N,\lambda,A,v_1,v_2,v_3]
\mapsto [\Sigma_+,-\Sigma_-,\Sigma_{13},\Sigma_{12},\Sigma_{23},-N,\lambda,A,v_1,v_3,v_2] $ \\
$\phi_3^{\pm}\! :~[\Sigma_+,\Sigma_-,\Sigma_{12},\Sigma_{13},\Sigma_{23},N,\lambda,A,v_1,v_2,v_3]
\mapsto [\Sigma_+,\Sigma_-,\pm \Sigma_{12},\mp \Sigma_{13},-\Sigma_{23},N,-\lambda,A,v_1,\pm v_2,\mp v_3]$\\
$\phi_4:~[\Sigma_+,\Sigma_-,\Sigma_{12},\Sigma_{13},\Sigma_{23},N,\lambda,A,v_1,v_2,v_3]
\mapsto[\Sigma_+,\Sigma_-,-\Sigma_{12},-\Sigma_{13},\Sigma_{23},N,\lambda,A,v_1,-v_2,-v_3]$
\end{tabular}$$
These discrete symmetries imply that without loss of  generality we can restrict the
variables $A\geq0$ and $N\geq0$,  since the dynamics in the other regions can be
obtained by simply applying one or more of  the maps above.  The third and fourth symmetries listed imply that one can add  additional constraints on the variables $\Sigma_{12},\Sigma_{13},v_2$ or $v_3$. For the type VI$_{-1/9}$ we can use $\phi_4$ to restrict $v_2$ to be non-negative (since $v_2=0$ implies $v_3=0$, see below); hence, we will assume $v_2\geq 0$. There is no natural way to restrict any of the remaining variables using the symmetry $\phi^+_3$, and so we will not do so here.

\subsection{Invariant sets}
For the case $h=-1/9$ we define ($N_{ab}v^av^b$ is identically zero for $h=-1/9$)
\begin{eqnarray}
\widehat{D}=\left[\lambda(\Sigma_{12}^2+\Sigma_{13}^2)+2\Sigma_{12}\Sigma_{13}\right].
\label{Dhat}\end{eqnarray}
In this analysis we will be concerned with the following invariant sets:
\begin{enumerate}
\item{} $T(VI_{-1/9})$: The general tilted type VI$_{-1/9}$  model: $ |\lambda|<1$.
\item{} $T_1(VI_{-1/9})$: A one-tilted type VI$_{-1/9}$ model: $|\lambda|<1$, $v_2=v_3=\Sigma_{12}=\Sigma_{13}=0$.
\item{} $T_{1,0}(VI_{-1/9})$: A one-tilted diagonal type VI$_{-1/9}$ model: $v_2=v_3=\Sigma_{12}=\Sigma_{13}=\Sigma_{23}=\lambda=0$.
\item{} $N^{\pm}(VI_{-1/9})$: A class of tilted type VI$_{-1/9}$ models:  $|\lambda|<1$, $\widehat{D}=0$.
\item{} $T_2^+(VI_{-1/9})$: A two-tilted type VI$_{-1/9}$ model. This is the fixed-point-set of $\phi_3^+$ and is given by  $v_3=\Sigma_{13}=\Sigma_{23}=\lambda=0$.
\item{} $T_2^-(VI_{-1/9})$: A one-tilted irrotational type VI$_{-1/9}$ model. This is the fixed-point-set of $\phi^-_3$ and is given by $v_2=v_3=\Sigma_{12}=\Sigma_{23}=\lambda=0$.
\item{} $B(VI_{-1/9})$: Non-tilted type VI$_{-1/9}$: $|\lambda|<1$, $v_1=v_2=v_3=\Sigma_{12}=\Sigma_{13}=0$.
\item{} $B_0(VI_{-1/9})$: A class of diagonal non-tilted type VI$_{-1/9}$ models ($n^{\alpha}_{~\alpha}=0$): $V=\Sigma_{12}=\Sigma_{13}=\Sigma_{23}=\lambda=0$
\item{} $T(II)$: The general type II model: $\lambda=\pm 1$, $A=0$.
\item{}$B(I)$: Type I: ${N}=A=V=0$.
\item{}{$\partial T(I)$}: ``Tilted'' vacuum type I: $\Omega=N=A=0$.
\end{enumerate}
Regarding $N^{\pm}(VI_{-1/9})$, to verify that this is indeed an invariant subspace, we calculate $\widehat{D}'$:
\[ \widehat{D}'=2\left(q-2-3\Sigma_+-\sqrt{3}\lambda\Sigma_{23}+3Av_1\right) \widehat{D};\]
hence, $\widehat{D}=0$ defines an invariant subspace.
Note also that this invariant set is not a manifold; it is similar to the light-cone in 2-dimensional Minkowski space. Therefore, $N^{\pm}(VI_{-1/9})-T_1(VI_{-1/9})$ consists of 4 disconnected pieces. By the symmetry $\phi_4$, these are actually only two inequivalent pieces. Here, we choose $N^{\pm}(VI_{-1/9})$ such that
\[ T_2^+(VI_{-1/9})\subset N^+(VI_{-1/9}), \quad T_2^-(VI_{-1/9})\subset N^-(VI_{-1/9}) \]
Since $N^+(VI_{-1/9})\cap N^-(VI_{-1/9})=T_1(VI_{-1/9})$, both $N^+(VI_{-1/9})$ and $N^-(VI_{-1/9})$ are invariant sets.

We note that the closure of the set $T(VI_{-1/9})$ is given by
\begin{eqnarray}
\overline{T(VI_{-1/9})}&=&T(VI_{-1/9})\cup T(II)\cup B(I) \cup \partial T(I).
\label{eq:decomp}\end{eqnarray}
Since the boundaries may play an important role in the evolution of the dynamical system  we must consider all of the sets in the decomposition (\ref{eq:decomp}).

Let us consider the constraint equations (\ref{const:v2}) and (\ref{const:v3}) as a
linear map \[ {\sf L}:~(\Sigma_{12},\Sigma_{13})\mapsto (v_2,v_3)/G_+,\] where ${\sf
L}$ is considered as given in terms of $A$, $N$, $\lambda$ and $\Omega$.  For $h\neq
-1/9$, $\det({\sf L})\neq 0$ and the image of ${\sf L}$ is 2-dimensional.  However, for
$h= -1/9$, $\det({\sf L})= 0$ and the image of ${\sf L}$ is 1-dimensional; hence, in
this sense \emph{the constraint equations are degenerate}.  This implies that
$(v_2,v_3)$ has to be restricted to a 1-dimensional submanifold.  We will therefore say
that the general type VI$_{-1/9}$ model only allows for 2 tilt degrees of freedom.  In
particular, we can solve for $v_3$ and obtain \[ v_3=-\frac{\lambda
v_2}{1+\sqrt{1-\lambda^2}}.\]

It is illustrative to consider the eigenvectors of the map ${\sf L}$:
\begin{eqnarray}
{\bf x}_0=\left(
-\frac{\lambda}{1+\sqrt{1-\lambda^2}}, ~ 1
\right),&& \quad {\sf L}{\bf x}_0=0, \nonumber \\
{\bf x}_a=\left(
1,~ -\frac{\lambda}{1+\sqrt{1-\lambda^2}}
\right),&& \quad {\sf L}{\bf x}_a=a{\bf x}_a, \quad a>0. \nonumber
\end{eqnarray}
For each of these eigenvectors, $\widehat{D}=0$ and hence, alternatively, we can define $N^{\pm}(VI_{-1/9})$  when $(\Sigma_{12},\Sigma_{13})$ is proportional to one of these eigenvectors. More specifically, for $N^+(VI_{-1/9})$, $(\Sigma_{12},\Sigma_{13})\propto {\bf x}_a$, while for $N^-(VI_{-1/9})$, $(\Sigma_{12},\Sigma_{13})\propto {\bf x}_0$.

Note also that, due to the presence of the eigenvector with zero eigenvalue ${\bf x}_0$, $(v_2,v_3)=0$ does not necessarily mean that $(\Sigma_{12},\Sigma_{13})$ is zero. In particular, for the non-tilted models this implies that we may have an additional shear degree of freedom; these models have usually been called the exceptional case and are denoted by an asterisk; e.g., $B_0(VI^*_{-1/9})$ and  $B(VI^*_{-1/9})$. We also note that for the tilted models, there is an exceptional case of the one-tilted models $T_1(VI_{-1/9})$ which could be denoted $T_1(VI^*_{-1/9})$. However, $T_1(VI^*_{-1/9})=N^-(VI_{-1/9})$ as explained above. Therefore, we keep the notation $N^-(VI_{-1/9})$. Similarly, we have $T_{1,0}(VI^*_{-1/9})=T_2^-(VI_{-1/9})$.

\begin{table}
\caption{The dimensions of the invariant sets for the Bianchi type $VI_{ -1/9}$
model. The right-most column indicates the specialization in terms of the $G_2$ cosmologies. Here, Diag means diagonal, HO means hypersurface orthogonal, and OT means orthogonally transitive. The stars indicate the exceptional cases where the models aquire an addition shear degree of freedom.}
\begin{tabular}{|c||cc|c|c|}
\hline
Dim &Invariant set & &No of Tilt & $G_2$ action \\
\hline
2 & $B_0(VI_{-1/9})$ &&0  & Diag \\ \hline
3 & $T_{1,0}(VI_{-1/9})$ & &1 & Diag \\
  & $B_0(VI^*_{-1/9})$& $\star$ &0 & HO \\
\hline
4 & $B(VI_{-1/9})$ & &0  & OT \\
  & $T_2^+(VI_{-1/9})$ & &2  & HO \\
  & $T_2^-(VI_{-1/9})$ & $\star$ &1  & HO \\
\hline
5 & $T_1(VI_{-1/9})$ & &1  & OT \\
  & $B(VI^*_{-1/9})$& $\star$ &0  & Gen $G_2$ \\
\hline
6 & $N^+(VI_{-1/9})$ & & 2  &Gen $G_2$ \\
  & $N^-(VI_{-1/9})$ &$\star$ & 1  &Gen $G_2$ \\
\hline
7 & $T(VI_{-1/9})$ & &2 & Gen $G_2$ \\
\hline
\end{tabular}\\
\end{table}
\subsection{Fluid Vorticity}
The various invariant subspaces can also be categorised in terms of
the ($H_{\mathrm{fluid}}$-normalised, where $H_{\mathrm{fluid}}\equiv (1/3)u^{\mu}_{~;\mu}$) fluid vorticity, $W^{\alpha}$. The vorticity of the fluid for the type VI$_{-1/9}$ models is given by:
\begin{eqnarray}
W_a=\frac{1}{2B}\left(N_{ab}+\varepsilon_{abc}A^c\right)v^b, \quad W_0=0,
\label{eq:vorticity}\end{eqnarray}
where
\[ B\equiv\frac{1-\frac 13(V^2+V^2\mathcal{S}+2A_av^a)}{[1-(\gamma-1)V^2]\sqrt{1-V^2}}.\]
For the invariant sets:
\begin{enumerate}
\item{} $T(VI_{-1/9})$: $W^0=W^1=0$, most general vortic type VI$_{-1/9}$.
\item{} $N^+(VI_{-1/9})$: $W^0=W^1=0$.
\item{} $N^-(VI_{-1/9})$: $W^0=W^a=0$, non-vortic.
\item{} $T_2^+(VI_{-1/9})$: $W^0=W^1=W^2=0$.
\item{} $T_2^-(VI_{-1/9})$: $W^0=W^a=0$, non-vortic.
\item{} $T_1(VI_{-1/9})$: $W^0=W^a=0$, non-vortic.
\item{} $B(VI_{-1/9})$: $W^0=W^a=0$, non-tilted and non-vortic.
\end{enumerate}
We note that the most general non-vortic model is of dimension 6. Hence, since the non-vortic type VI$_h$ and VII$_h$ models -- regarding $h$ as fixed -- are of dimension 5, the type VI$_{-1/9}$ model is the most general non-vortic model of all Bianchi models.

In general we can also solve for the vorticity component $W^2$:
\[ W^2=\frac{\lambda}{1+\sqrt{1-\lambda^2}}W^3.\]
This follows from the constraint equations and eq.(\ref{eq:vorticity}).

\section{Qualitative behaviour}
\label{sect:Qual}
\subsection{Monotone functions}
There are a number of monotone functions in the state space of interest. For $0<\gamma\leq 6/7$, there exists a monotonically increasing function $Z_1$ defined by
\begin{eqnarray}
Z_1&\equiv& \alpha\Omega^{1-\gamma}, \quad \alpha=\frac{(1-V^2)^{\frac 12(2-\gamma)}}{G^{1-\gamma}_+V^{\gamma}_{\phantom{+}}}, \\
Z_1'&=&\left[2(1-\gamma)q+(2-\gamma)+{\gamma}\mathcal{S}\right]Z_1.\nonumber
\end{eqnarray}
This function can be used to show \cite{CH2}:
\begin{thm}\label{thm:nontilted}
For $0<\gamma\leq 6/7$, all tilted Bianchi models  (with $\Omega>0$, $V<1$) of solvable type are asymptotically non-tilted at late times.
\end{thm}
\begin{cor}[Cosmic no-hair]
For $\Omega>0$, $V<1$, and $0<\gamma<2/3$ we have that
\[ \lim_{\tau\rightarrow \infty}\Omega=1, \quad \lim_{\tau\rightarrow \infty}V=0.\]
\label{no-hair}
\end{cor}

Moreover, the following function is a monotone function in $T(VI_{-1/9})$:
\begin{eqnarray}
Z_2&=&\frac{A^4N^2G_+^5\widehat{D}^2}{(1-V^2)^{\frac 52(2-\gamma)}\Omega^5}, \\
Z_2'&=&(5\gamma-6)(3-2Av_1)Z_2,
\end{eqnarray}
where $\widehat{D}$ is defined in eq.(\ref{Dhat}).
This function is monotonically decreasing for $\gamma<6/5$ and monotonically increasing for $6/5<\gamma$.

We note that in the subspace $N^-(VI_{-1/9})$ we have the monotone function:
\begin{eqnarray}
Z_3&=&\frac{v_1^2\Omega}{A^2G_+(1-V^2)^{\frac 12(2-\gamma)}}, \\
Z_3'&=&-(2-\gamma)(3-2Av_1)Z_3.
\end{eqnarray}
This function is monotonically decreasing in $N^-(VI_{-1/9})$.

The monotonic function $Z_3$ immediately implies:
\begin{thm}[Future behaviour in $N^-(VI_{-1/9})$]
For $2/3<\gamma<2$, $\Omega>0$, $A>0$, $v_1^2<1$, $v_2=v_3=0$ we have that:
\[ \text{either}\quad \lim_{\tau\rightarrow \infty}\Omega=0, \quad
\text{or}\quad  \lim_{\tau\rightarrow \infty}V=0.\]
\end{thm}
This implies that all irrotational  type VI$_{-1/9}$ universes are either asymptotically vacuum or non-tilted at late times.
\subsection{Equilibrium points}

\subsubsection{$B(I)$: Equilibrium points of Bianchi type I}
\begin{enumerate}
\item{}$\mathcal{I}(I)$: $\Sigma_+=\Sigma_-=\Sigma_{12}=\Sigma_{13}=\Sigma_{23}=A=N=V=0$ and $\Omega=1$. Here, $|\lambda|<1$ and is an unphysical parameter.  This represents the flat Friedman-Lema{\^i}tre model.
\end{enumerate}
The remaining equilibrium points are all in $\partial T(I)$.

\subsubsection{$T(II)$: Equilibrium points of Bianchi type II}
All of the tilted equilibrium points come in pairs. These represent identical solutions (they differ by a frame rotation); however, since their embeddings in the full state space are inequivalent, two of their eigenvalues are different. All of these equilibrium points have an unstable direction with eigenvalue $-2\sqrt{3}\Sigma_{23}$ corresponding to the variable $A$.
These equilibrium points are given in \cite{HHC}.

\subsubsection{$T(VI_{-1/9})$:  Equilibrium points of Bianchi type VI$_{-1/9}$}
\begin{enumerate} \item{} $\mathcal{C}(-1/9)$:  Collins perfect fluid solution,
$2/3<\gamma<5/3$ \\ $\Sigma_{12}=\Sigma_{13}=\Sigma_{23}=\lambda=V=0$, $\Sigma_+=-\frac
1{4}(3\gamma-2)$, $\Sigma_-=\frac{\sqrt{3}}{12}(3\gamma-2)$, $N^2=\frac
3{16}(3\gamma-2)(2-\gamma)$, $A=N/\sqrt{3}$, $\Omega=\frac 13(5-3\gamma)$.  This
equilibrium point is in $B(VI_{-1/9})$.  \item{} $\mathcal{R}^+(-1/9)$:  The
Apostolopoulos $h=-1/9$ solution \cite{Apo2}, $4/3<\gamma<3/2$\\ This is a vortic
solution lying in the invariant subspace $T^+(VI_{-1/9})$.  The solution is given in
terms of the expansion-normalised variables in \cite{HHLC2} with $h=-1/9$, $k=1/3$.
\item{} Bianchi type VI$_{-1/9}$ vacuum plane waves.  All of these solutions have \[
\Omega=\Sigma_{12}=\Sigma_{13}=\Sigma_{23}=0, ~\Sigma_-=N=\sqrt{-\Sigma_+(1+\Sigma_+)},
~A=(1+\Sigma_+),~ -1<\Sigma_+<-3/4,~ |\lambda|<1.\] It is avantageous to introduce
$r\equiv\sqrt{1-\lambda^2}$, which implies that we can write \[
\Sigma_+=-\frac{3}{3+r^2}, \quad 0<r\leq 1.  \] We will also define $\rho$ by \[
\rho=v_2^2+v_3^2.\]

The equilibrium points are then determined by the tilt velocities:  \begin{enumerate}
\item{} $\mathcal{L}(-1/9)$:  $v_1=v_2=v_3=0$.  These represent 'non-tilted' plane
waves and lie in $B(VI_{-1/9})$.  \item{} $\widetilde{\mathcal{L}}(-1/9)$:
$v_1=\frac{3\gamma(3+r^2)-2(9+2r^2)}{2r^2(\gamma-1)}$, $v_2=v_3=0$,
$\frac{6(3+r^2)}{9+5r^2}<\gamma<2$.  These represent 'intermediately tilted' plane
waves and lie in $T_1(VI_{-1/9})$.  \item{} $\widetilde{\mathcal{L}}_{\pm}(-1/9)$:
$v_1=\pm 1$, $v_2=v_3=0$.  These represent 'extremely tilted' plane waves and lie in
$T_1(VI_{-1/9})$.  \item{} \label{defF} $\widetilde{\mathcal{F}}^+(h)$:  Here,
$(9+7r^2)/[3(3+r^2)]\leq \gamma\leq 3(3+r^2)/(9-r^2)$ and \begin{eqnarray}
v_1&=&-\frac{3\gamma(3+r^2)-(9+7r^2)}{2r^2(3-\gamma)},\quad v_2^2-v_3^2= \rho r
\nonumber \\
\rho&=&\frac{(9+r^2)\left[5-3\gamma\right]\left[3\gamma(3+r^2)-(9+7r^2)\right]}{8r^4(3-\gamma)^2}
\nonumber \end{eqnarray} These represent 'intermediately tilted' plane waves and lie in
$N^+(VI_{-1/9})$ (for $\lambda=0$ they lie in $T_2^+(VI_{-1/9})$).  \item{}
$\widetilde{\mathcal{E}}_p^+(-1/9)$, $0<\gamma<2$:  \begin{eqnarray}
v_1&=&-\frac{4r^2}{3(3-r^2)},\quad v_2^2-v_3^2=\rho r \nonumber \\ \rho&=&
1-v_1^2=\frac{(9+r^2)(9-7r^2)}{9(3-r^2)^2} \nonumber \end{eqnarray} These represent 'extremely
tilted' plane waves and lie in $N^+(VI_{-1/9})$.  For $\lambda=0$ these equilibrium
points lie in $T_2^+(VI_{-1/9})$, and due to the special importance for the late-time
behaviour, we will denote $\widetilde{\mathcal{E}}_{p0}^+(-1/9)\equiv
\left.\widetilde{\mathcal{E}}_p^+(-1/9)\right|_{\lambda=0}$.  Therefore, \[
\widetilde{\mathcal{E}}_{p0}^+(-1/9):  ~ v_1=-\frac 23, \quad v_2=\frac{\sqrt{5}}{3}.\]
\end{enumerate} \item{} $\mathcal{CF}$:  The Collinson-French (Robinson-Trautmann)
solution is given by:  \begin{eqnarray} &&\Sigma_+=-\frac 13,\quad \Sigma_-=\frac{1}{3\sqrt{3}},
\quad \Sigma_{13}=\frac{\sqrt{15}}{9},\quad N=\frac{1}{\sqrt{2}}, \quad
A=\frac{1}{\sqrt{6}},\nonumber \\ && \Sigma_{12}=\Sigma_{23}=\Omega=\lambda=0.
\nonumber \end{eqnarray} There are the following equilibrium points associated with the
Collinson-French solution:  \begin{enumerate} \item{}$\mathcal{CF}_0$:  $v_1=v_2=0$,
$0<\gamma<2$.  \item{}$\widetilde{\mathcal{CF}}_{1+}$:
$v_1=-\frac{\sqrt{6}(3\gamma-4)}{2(3-\gamma)}$,
$v_2=\frac{\sqrt{5(3\gamma-4)(3-2\gamma)}}{\sqrt{2}(3-\gamma)}$, $\frac 43<\gamma<\frac
32$.  \item{} $\widetilde{\mathcal{CF}}_{2}$:
$v_1=\frac{\sqrt{6}(9\gamma-14)}{6(\gamma-1)}$, $v_2=0$,
$\frac{24-\sqrt{6}}{15}<\gamma<\frac{24+\sqrt{6}}{15}$.
\item{}$\widetilde{\mathcal{ECF}}_{\pm}$:  $v_1=\pm 1$, $v_2=0$, $0<\gamma<2$.
\end{enumerate} These equilibrium points, and their stability, were studied in
\cite{CH2}.

\item{} $\mathcal{W}$: Wainwright $\gamma=10/9$ solution:
\begin{eqnarray}
&&\Sigma_+=-\frac 13,\quad \Sigma_-=\frac{1}{3\sqrt{3}},\quad 0<\Sigma_{13}<\frac{\sqrt{15}}{9},\quad
N=\frac{1}{6}\sqrt{8+54\Sigma_{13}^2}, \quad A=\frac{1}{\sqrt{3}}N,\quad \Omega=\frac 59-3\Sigma_{13}^2,\nonumber \\
&& \Sigma_{12}=\Sigma_{23}=\lambda=V=0.  \nonumber
\end{eqnarray}
\end{enumerate}

\section{Late-time behaviour}

The late-time behaviour of models with $0<\gamma<2/3$ is determined by
(the Cosmic no-hair) Corollary 3.2.

\subsection{The invariant subspace $N^+(VI_{-1/9})$}
Here, we have the following late-time attractors:
\begin{itemize}
\item{} $2/3<\gamma\leq 4/3$: The Collins solution, $\mathcal{C}(-1/9)$.
\item{} $4/3<\gamma<3/2$: The Apostolopoulos solution, $\mathcal{R}^+(-1/9)$.
\item{} $3/2\leq \gamma<2$: "Extremely tilted" vacuum plane waves, $\widetilde{\mathcal{E}}^+_{p0}({-1/9})$
\end{itemize}
The stability of these points for $\gamma<3/2$ follows from the eigenvalues of the linearised matrix. For $3/2\leq \gamma<2$  several zero-eigenvalues occur and a centre manifold analysis is needed to determine
the late-time behaviour.
\subsubsection{The case $3/2 <\gamma<2$: the centre manifold}   Let us present the
centre manifold analysis of the equilibrium point $\widetilde{\mathcal{E}}^+_{p0}({-1/9})$  in some detail.
The centre manifold in this case is a 2-dimensional submanifold of the 5-dimensional extremely tilted invariant subspace $\left.N^+(VI_{-1/9})\right|_{V=1}$. To find the centre manifold, we will therefore set $V=1$. Let us choose variables
\begin{eqnarray}
(\Sigma_+,\Sigma_{23},N,\lambda,v_2)=\left(-\frac 34+x_1,x_2,\frac{\sqrt{3}}{4}+x_3,x_4,\frac{\sqrt{5}}3+x_5\right),
\end{eqnarray}
let $\Sigma_-$, $\Sigma_{12}$, $\Sigma_{13}$ and $\Omega$ be determined from the constraint equations, and $v_1^2=1-v_2^2-v_3^2$. Let us define the column vector ${\sf x}=[x_1,x_2,x_3,x_4,x_5]^T$. We can now expand the equations of motion to 2nd order in ${\sf x}$:
\begin{eqnarray}
{\sf x}'={\sf A}{\sf x}+{\sf C}({\sf x},{\sf x})+\mathcal{O}({\sf x}^3),
\end{eqnarray}
where ${\sf C}(-,-)$ is a bilinear vector-valued function.
It is convenient to align the vector ${\sf x}$ with the Jordan canonical form of ${\sf A}$. This can be accomplished by defining
\begin{eqnarray}
{\sf P}=\begin{bmatrix}
\frac 75 & -\frac 25 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 \\
\frac{2\sqrt{3}}5 & -\frac{2\sqrt{3}}5 & 0 & 0 & 0 \\
4\sqrt{3} & -4\sqrt{3} & 0 & 4\sqrt{3} & -4\sqrt{3} \\
-\frac{48\sqrt{5}}{25} & \frac{64\sqrt{5}}{45} & \frac{112\sqrt{5}}{225} & 0 & 0
\end{bmatrix}.
\end{eqnarray}
Now, defining ${\sf y}={\sf P}^{-1}{\sf x}$, the equation for ${\sf y}$ becomes
\[
{\sf y}'={\sf J}{\sf y}+\tilde{{\sf C}}({\sf y},{\sf y})+\mathcal{O}({\sf y}^3),
\]
where ${\sf J}$ is the Jordan block matrix
\[ {\sf J}=\mathrm{diag}\left(0,-\frac 12, -\frac 56, 0, -\frac 12\right).\]
The centre manifold correspond to the zero-eigenvalues of ${\sf J}$.
We can therefore parameterise the centre manifold using the variables $(y_1,y_4)$.
The next step is to expand the variables $y_i$, $i=2,3,5$, in terms of $(y_1,y_4)$ on
the centre manifold, to second order. We therefore define the quadratic forms $Y_i(y_1,y_4)$, $i=2,3,5$, such that $y_i-Y_i(y_1,y_4)=0$ is (to second order) an invariant subspace. On the centre manifold we then have:
\begin{eqnarray}
y_i=Y_i(y_1,y_4)+\mathcal{O}({\sf y}^3).
\end{eqnarray}
By substituting these into the $y_1$ and $y_4$ evolution equations, we finally get, on the centre manifold:
\begin{eqnarray}
y_1'&=&-16y_1^2+\mathcal{O}({\sf y}^3), \nonumber \\
y_4'&=&-16y_1y_4+\mathcal{O}({\sf y}^3).
\end{eqnarray}
To lowest order, these equations can be solved to give $y_1\approx 1/(16\tau)$,
$y_4\approx C/\tau$, where a constant of integration has been eliminated by a
translation of time.  Note that, in principle, there are essentially two kinds of behaviour:  for
$\tau<0$ the variables diverge, while for $\tau>0$ the variables decay.
However, requiring $\Omega>0$ leaves only the decaying mode as physically acceptable.
These decay rates will therefore be the dominant ones since the centre manifold will
dominate the behaviour at late times.

In terms of the original variables, we therefore get the decay rates in $N^+(VI_{-1/9})$ (only dominant decay rates are included):
\begin{eqnarray}
\Sigma_+&\approx & -\frac 34\left(1-\frac{7}{60\tau}\right), \nonumber \\
\Sigma_-&\approx & \frac{\sqrt{3}}4\left(1+\frac{1}{60\tau}\right), \nonumber \\
\Sigma_{12}&\approx & \frac{\sqrt{15}}{60\tau}, \nonumber \\
\Sigma_{13}&\approx & -\frac{(1+16C)\sqrt{5}}{160\tau^2}, \nonumber \\
\Sigma_{23}&\approx & -\frac{(1+16C)}{8\tau^2}, \nonumber \\
N&\approx & \frac{\sqrt{3}}4\left(1+\frac{1}{10\tau}\right), \nonumber \\
\lambda &\approx & \frac{(1+16C)\sqrt{3}}{4\tau}, \nonumber \\
\Omega &\approx & \frac{3}{40\tau}, \nonumber \\
v_1&\approx & -\frac{2}3\left(1+\frac{9}{20\tau}\right), \nonumber \\
v_2&\approx & \frac{\sqrt{5}}3\left(1-\frac{9}{25\tau}\right), \nonumber \\
\sqrt{1-V^2}&\approx & (\sqrt{1-V^2})_0\exp\left[-\frac{5(2\gamma-3)}{3(2-\gamma)}\tau\right].
\end{eqnarray}
This confirms that the 'extremely tilted' vacuum plane wave  $\widetilde{\mathcal{E}}^+_{p0}(-1/9)$ is the attractor for $N^+(VI_{-1/9})$, $3/2<\gamma<2$. Note, however, that this solution is unstable in the fully tilted space $T(VI_{-1/9})$, which can be verified by calculating, for example, $\widehat{D}'\approx (3/4)\widehat{D}$ close to $\widetilde{\mathcal{E}}^+_{p0}(-1/9)$.

\subsection{The invariant subspace $N^-(VI_{-1/9})$: non-vortic type VI$_{-1/9}$ models}
\begin{itemize}
\item{} $2/3<\gamma<10/9$: The Collins solution, $\mathcal{C}(-1/9)$.
\item{} $\gamma=10/9$: The Wainwright solution, $\mathcal{W}$.
\item{} $10/9<\gamma<2$: The Collinson-French solution,
$\widetilde{\mathcal{C}\mathcal{F}}$. We have the following refinement for the
asymptotic tilt (which follows from an analysis of the eigenvalues):
\begin{itemize}
\item[$\star$] $10/9<\gamma\leq (24-\sqrt{6})/15$: The tilt is asymptotically zero ($\mathcal{C}\mathcal{F}_0$).
\item[$\star$] $(24-\sqrt{6})/15<\gamma\leq 14/9$: The tilt is either asymptotically zero ($\mathcal{C}\mathcal{F}_0$) or extreme ($\widetilde{\mathcal{E}\mathcal{C}\mathcal{F}}_-$).
\item[$\star$] $14/9<\gamma<(24+\sqrt{6})/15$: The tilt is either asymptotically intermediate ($\widetilde{\mathcal{C}\mathcal{F}}_2$) or extreme ($\widetilde{\mathcal{E}\mathcal{C}\mathcal{F}}_-$).
\item[$\star$] $(24+\sqrt{6})/15\leq \gamma<2$: The tilt is asymptotically extreme
($\widetilde{\mathcal{E}\mathcal{C}\mathcal{F}}_-$).
\end{itemize}
\end{itemize}

\subsection{$T(VI_{-1/9})$: The fully tilted type VI$_{-1/9}$ model}
For the fully tilted model, the late-time behaviour can be summarised as follows:
\begin{itemize}
\item{} $2/3<\gamma<10/9$: The Collins solution, $\mathcal{C}(-1/9)$.
\item{} $\gamma=10/9$: The Wainwright solution, $\mathcal{W}$.
\item{} $10/9<\gamma<2$: The Collinson-French solution, $\widetilde{\mathcal{C}\mathcal{F}}$. We have the following refinement for the asymptotic tilt:
\begin{itemize}
\item[$\star$] $10/9<\gamma\leq (24-\sqrt{6})/15$: The tilt is asymptotically zero  ($\mathcal{C}\mathcal{F}_0$).
\item[$\star$] $(24-\sqrt{6})/15<\gamma\leq 4/3$: The tilt is either asymptotically zero  ($\mathcal{C}\mathcal{F}_0$) or extreme ($\widetilde{\mathcal{E}\mathcal{C}\mathcal{F}}_-$).
\item[$\star$] $4/3<\gamma\leq 5/6+\sqrt{721}/42$: The tilt is either asymptotically intermediate ($\widetilde{\mathcal{C}\mathcal{F}}_{1+}$)  or extreme ($\widetilde{\mathcal{E}\mathcal{C}\mathcal{F}}_-$).
\item[$\star$] $5/6+\sqrt{721}/42<\gamma<\gamma_H\approx 1.47392$: The tilt
is either asymptotically oscillatory (closed curve) or extreme ($\widetilde{\mathcal{E}\mathcal{C}\mathcal{F}}_-$).
\item[$\star$] $\gamma_H<\gamma<2$: The tilt is asymptotically extreme ($\widetilde{\mathcal{E}\mathcal{C}\mathcal{F}}_-$).
\end{itemize}
\end{itemize}
Most of these results comes from an analysis of the eigenvalues of the various
equilibrium points. However, for $\gamma= 5/6+\sqrt{721}/42$ the equilibrium point
$\widetilde{\mathcal{C}\mathcal{F}}_{1+}$ undergoes a Hopf-bifurcation and
a stable closed curve results. The analysis of this Hopf-bifurcation is given below. Interestingly, this closed curve co-exists with an extremely tilted attractor.

\subsubsection{The Hopf-bifurcation}
Our main aim for this  section is to show the following.
\begin{thm}
There exists a $\gamma_0$ such that for $\frac 56+\frac{\sqrt{721}}{42}<\gamma<\gamma_0$  there exists a closed orbit, $c(\tau)$, acting as the future attractor for a set of non-zero measure of tilted Bianchi type VI$_{-1/9}$ models.
\end{thm}
To prove this theorem we will first show the existence of a closed period orbit
acting as an attractor in a particular subset. We consider the invariant subset given by the Collinson-French solution with 2 tilts. We introduce $(X,Y)=(v_1,v_2^2)$:
\begin{eqnarray}
X'&=&\left(T-\frac 23\right)X-\frac{2\sqrt{6}}{3}Y, \nonumber \\
Y'&=&2\left(T+\frac {2\sqrt{6}}3X\right)Y.
\label{2dsys}\end{eqnarray}
We set $X_0=-\frac{\sqrt{6}(3\gamma-4)}{2(3-\gamma)}$,
$Y_0=\frac{5(3\gamma-4)(3-2\gamma)}{2(3-\gamma)^2}$, and perform the
transformation $x=X-X_0$, $y=Y-Y_0$ with respect to the equilibrium point $\widetilde{\mathcal{C}\mathcal{F}}_{1+}$.

The normal form of a Hopf-bifurcation can be written
\begin{eqnarray} Z'=(\lambda+b|Z|^2)Z, \end{eqnarray}
where $b$ is some complex number and $\lambda=\alpha+i \beta$ is a parameter.
If $\mathrm{Re}(b)<0$ for $\alpha=0$, there exists a stable closed orbit for
$0<\alpha$ sufficently small.

We will therefore set $\gamma=\frac 56+\frac{\sqrt{721}}{42}$ and expand to cubic terms in $x$ and $y$. It is also convenient to introduce a complex variable $z$ chosen such that it aligns with the Jordan form of the linearised matrix. This can be achieved by setting:
\[ z=x+iay,\]
where $a$ is a real number chosen such that the linear term of $z'=f(z,\bar{z})$ is
\begin{eqnarray}
\partial_zf(0,0)=\frac{i}{6}\sqrt{-1205+45\sqrt{721}}, \quad  \partial_{\bar{z}}f(0,0)=0.
\end{eqnarray}
By a transformation
\[ Z=z+a_{11}z^2+a_{12}z\bar{z}+a_{22}\bar{z}^2+a_{111}z^3+a_{112}z^2\bar{z}+a_{122}z\bar{z}^2+a_{222}\bar{z}^3, \]
 we can choose the coefficients $a_{ij}$ and $a_{ijk}$ so that the equation for $Z$ takes the form
\begin{eqnarray} Z'=(\lambda+b|Z|^2)Z +\mathcal{O}(|Z|^4),
\end{eqnarray}
where
\[ \lambda=\frac{i}{6}\sqrt{-1205+45\sqrt{721}},\quad \mathrm{Re}(b)=\frac{1}{291600}\left(-15990233+595193\sqrt{721}\right)\approx -0.029.\]
Hence, there exists a $\gamma_0$ such that there exists a closed stable orbit for $\frac 56+\frac{\sqrt{721}}{42}<\gamma<\gamma_0$.

The next step is to show that this orbit is also stable in the fully tilted type
VI$_{-1/9}$ models. We can show this as follows: for a function $B$, we introduce
\emph{the average}, $\av{B}$, with respect to the closed orbit $c(\tau)$, defined by
\[ \av{B}=\frac{1}{T}\oint_{c(\tau)}B\d \tau, \quad T=\oint_{c(\tau)}\d\tau.\]
We can use this average, using similar manipulations as for the closed curves in the type IV, VI$_h$ and VII$_h$ models \cite{HHC,HHLC2} to show that:
\begin{thm}
Assume that there exists a closed periodic orbit $c(\tau)$ for the dynamical system (\ref{2dsys}). Then
\[ \av{X}=-\frac{\sqrt{6}(3\gamma-4)}{2(3-\gamma)}, \quad \av{\lambda_{\Omega}}=-\frac{5(5\gamma-6)}{3(3-\gamma)}.\]
\end{thm}
\begin{proof}
From the $Y$ equation, we get $\av{T}=-2\sqrt{6}\av{X}/3$. A manipulation of the $V$
equation yields $\av{T}=\av{\mathcal{S}}$, and
$\av{\mathcal{S}}=(3\gamma-4)-2(\gamma-1)A\av{X}$.
These can now be solved to yield the desired value for $\av{X}$.
A similar manipulation of the
$\Omega$ equation yields $\av{\lambda_{\Omega}}$.
\end{proof}
This implies that the vacuum solution is stable when the closed curve is
perturbed by $\Omega$. Hence, since the Collinson-French solution is
stable with respect to vacuum perturbations, this closed curve is stable for
the fully tilted type VI$_{-1/9}$ models whenever
$\frac 56+\frac{\sqrt{721}}{42}<\gamma<\gamma_0$.

It remains to determine the maximal value for $\gamma_0$? It seems that this
limiting value is related to the existence of a heteroclinic orbit originating and ending at the
saddle points $\mathcal{C}\mathcal{F}_0$ and  $\widetilde{\mathcal{C}\mathcal{F}}_2$, respectively.
For the limiting value of $\gamma$, which we will call $\gamma_H$, there exists
such a heteroclinic orbit, while for values $\gamma\neq \gamma_H$ no such
heteroclinic orbit exists connecting these two equilibrium points.
We can use this to numerically estimate the value for $\gamma_H$. Our estimate gives:
\[ 1.473920<\gamma_H<1.473921.\]
Since $5/6+\sqrt{721}/42=1.472653409...$ this means that the region in which the
closed orbit exists is extremely small.\footnote{However, this seems to be
typical for these models, the loophole for the type IV and VII$_h$ models also
appears to be extremely small.}

Note also that this implies that these closed orbits co-exist with the extremely
tilted attractor $\widetilde{\mathcal{E}\mathcal{C}\mathcal{F}}_-$, which makes
this closed orbit even more difficult to detect. In addition, it appears as if the
curves asymptote to the closed curve relatively slowly, which implies that the
numerics have to run for a relatively long time in order to see the late-time asymptote. A numerical plot of some generally tilted type VI$_{-1/9}$ models approaching this closed orbit is shown in Fig.\ref{FigCC}.
\begin{figure}[f]
\caption{Bianchi type VI$_{-1/9}$ universes  approaching a closed curve and the equilibrium point $\widetilde{\mathcal{E}\mathcal{C}\mathcal{F}}_-$ ($\gamma=1.4735$). }\label{FigCC}
$$\includegraphics[scale=0.5,angle=90]{sixesninthed2.eps}$$
\end{figure}

\section{Conclusion}
In this paper we have examined the tilted Bianchi type VI$_{-1/9}$ model
in some detail.
This model is a special case (due to the vanishing of one of the constraint equations)
and necessitates a separate analysis from the general type
VI$_h$ models.  We showed that in these models there exists a tiny region of
parameter space where there exists a closed curve acting as an attractor.  This closed
curve co-exists with an extremely tilted attractor.
In the case of the most general irrotational models this closed curve,
which appears in terms of the tilt
velocities, is absent.
We have also confirmed the
analytical results with an extensive numerical investigation.

\newpage


\section{Introduction}

Cosmological and astrophysical observations provide growing evidence
of the existence of Dark Matter (DM) in our Universe. One of the
favored candidates is a weakly interacting massive particle (WIMP), a
new stable neutral particle with typical mass in the 10~GeV--TeV range
which interacts weakly~\cite{reviews}. These particles arise naturally
in various extensions of the Standard Model of Particle Interactions
(SM), e.g. the lightest neutralino in supersymmetric models with
R-parity conservation~\cite{reviews}, stable scalars in little Higgs
models~\cite{littleHiggs}, the lightest Kaluza-Klein excitation of the
SM fields in models with universal
extra-dimensions~\cite{extraD}. Direct searches for WIMPs look for the
recoil energy of nuclei due to interactions with the DM particles
passing through the Earth~\cite{directDM}. WIMP signatures can also be
found indirectly, namely by observing gamma-rays, positrons,
anti-protons and neutrinos produced in dark matter
annihilations~\cite{reviews}. As the annihilation rate scales with the
square of the dark matter density, indirect signals are expected to be
stronger where DM accumulates, as, for instance, the galactic center
and astrophysical bodies. High-energy neutrinos are produced in the
annihilation either directly or by subsequent decay of SM fermions and
bosons~\cite{DMnus} and their spectrum could provide information on
the WIMP mass, the WIMP-proton cross section and the branching ratios
of the various annihilation channels, and hence, on the DM properties.

Searches for high energy neutrinos can be performed in \v{C}erenkov
neutrino detectors/telescopes~\cite{sklimit,nutelescopes}, as
Super-Kamiokande (SK), AMANDA, IceCube, but no positive evidence has
been found so far. However, these detectors are unable to fully
reconstruct the neutrino energy and therefore cannot provide
information on the neutrino spectrum. In the case of small detectors
with large photo-multiplier coverage, like SK, the size limits the
maximum energy for which the events are contained, so only
through-going muons are of interest for energies of tens of GeV or
higher. In the case of neutrino telescopes like IceCube, least
granular, the size limits the energy threshold for detection. In both
cases and for the  energies of interest, these detectors can only
count neutrinos above their energy threshold. On the other hand,
different conceptual designs for detectors for future neutrino long
baseline experiments show that Magnetized Iron Calorimeter Detectors
(MIND)~\cite{MIND}, or its variants as MONOLITH~\cite{monolith} or
INO~\cite{ino}, Liquid Argon Time Projection Chambers
(LArTPC)~\cite{LAr} and Totally Active Scintillator Detectors
(TASD)~\cite{TASD} are able to measure the energy and direction of the
initial neutrino with good precision at energies of tens of GeVs.

In the present letter, we restrict our analysis to MIND, which are
conceptually similar to the existing MINOS detector~\cite{MINOS} but
with a mass one order of magnitude larger. Due to its ability for a
precise muon charge discrimination, they have been proposed as
ideal detectors for future neutrino factories. For the same reason,
these detectors have also been considered as atmospheric neutrino
observatories, opening up the possibility of determining the neutrino
mass hierarchy and the small mixing angle $\theta_{13}$, due to matter
effects in the oscillations of upward-going atmospheric
neutrinos~\cite{atmos}. Here, we reveal a novel aspect of
MIND~\cite{LArDM}, showing that they might constitute excellent
observatories to detect neutrinos from WIMP annihilations. For a given
value of the WIMP mass, we study the neutrino signal from WIMP
annihilations in these detectors, and exploit their energy and
angular resolution in order to reconstruct the neutrino spectrum and,
ultimately, to study DM properties~\cite{BEK97}. By measuring the
branching ratios of annihilation into different channels and the
WIMP-proton cross section, these measurements could provide a unique
opportunity for getting information on the WIMP couplings with the
particles of the SM.

\section{Neutrinos from WIMPs annihilations}

When a WIMP goes through a celestial body, as the Sun, it might
interact elastically with the nuclei and get scattered to a velocity
smaller than the escape velocity, remaining gravitationally trapped in
the body. It will then undergo additional scatterings, settling in the
Sun core, giving raise to an isothermal distribution. The WIMPs so
accumulated can annihilate with other WIMPs into SM particles, as
quarks, leptons and, if kinematically allowed, gauge and Higgs
bosons. Typically in the Sun, for sufficiently high capture rate and
annihilation cross section, equilibrium is reached and the
annihilation rate \mbox{$\Gamma_{\mathrm{ann}} \;$} is related to the capture rate $C_\odot$
as $\Gamma_{\mathrm{ann}} = 1/2 C_\odot$, which, for the Sun, is given
by~\cite{Gould92}
\begin{eqnarray}
C_\odot & \simeq & 9 \times 10^{24} \; {\rm s}^{-1} \,
\left(\frac{\rho_{\rm local}}{0.3 \,
{\rm GeV/cm}^{3}}\right) \, \left(\frac{270 \, {\rm
    km/s}}{\bar{v}_{\rm local}}\right)^3 \nonumber \\
& & \times \left(\frac{\sigma}{10^{-2} \, {\rm pb}}\right)
    \; \left(\frac{50 \, {\rm GeV}}{m_{\rm DM}}\right)^2 ~,
\end{eqnarray}
where $\rho_{\rm local}$ is the local WIMPs density, $\bar{v}_{\rm
local}$ is the velocity dispersion of WIMPs in the halo, $\sigma$ is
the WIMP-proton cross section, and $\mbox{$m_{\mbox{}_\mathrm{DM}}$}~$ is the WIMP mass. The
capture rate can be different by several orders of  magnitude for
spin-dependent and spin-independent interactions. Typically, for most
neutralino models as well as for Kaluza-Klein DM, the spin-dependent
cross-section dominates over the spin-independent one. The opposite
happens for scalar WIMPs. These scattering cross sections are
constrained in direct DM searches~\cite{directDMbounds}, which put
very strong bounds on the spin-independent ones. Since, in the Earth,
the abundance of nuclei with odd mass number is extremely small, the
neutrino flux from WIMP annihilations is strongly constrained by the
bounds on the spin-independent cross section and would be too low to
be interesting for detection in MIND, LArTPC or TASD. Thus, for our
purposes we will focus on WIMP annihilations in the Sun, for which the
much more weakly constrained spin-dependent cross section may play the
dominant role.

High-energy neutrinos would be produced directly or indirectly via the
decay of other products of the annihilations~\cite{reviews}. A broad
spectrum of neutrinos is so generated and depends on the WIMP mass and
on the branching ratios into the various channels:
\begin{equation}
\frac{dN_\nu}{d\Omega\ dt\ dE_\nu}=\frac{\Gamma_{\mathrm{ann}}}{4\pi
  R^2}\sum_i {\rm BR}_i \frac{dN_{i}}{dE_\nu}~,
\label{eq:fluxes}
\end{equation}
where the sum includes the possible annihilation channels with
spectrum $dN_i/dE_\nu$ and branching ratio ${\rm BR}_i$, and $R$ is
the Sun-Earth distance. If the DM is a Majorana fermion, the
annihilation amplitude into fermion pairs is proportional to the
fermion mass, so that the neutrino channel is irrelevant and
annihilations are dominated by heavy fermions, $b$s, $\tau$s, $c$s
and, if kinematically allowed, $t$s. Neutralinos can annihilate into
fermions as well as gauge bosons and Higgs and the dominant channels
are controlled by the neutralino composition~\cite{reviews}.
Typically, for neutralinos lighter than the W boson, the annihilation
into $b\bar{b}$ pairs gives the main contribution, with harder
neutrinos from $\tau \bar{\tau}$. For higher masses, the gauge bosons
channels are allowed and can have the highest branching ratio.
Kaluza-Klein WIMPs can annihilate directly into
neutrinos~\cite{extraD} with a branching ratio of few per cent and
would provide a specific signature with a peak in the neutrino
spectrum at $E_\nu=\mbox{$m_{\mbox{}_\mathrm{DM}}$}$. Their main annihilation channels are charged
leptons and light quarks with the neutrino flux generated dominantly
by the $\tau \bar{\tau}$ mode. In the following, we will treat the
branching ratios as free parameters as their exact values depend on
the type of particle considered and on its couplings.

Annihilations into $\tau$'s and gauge bosons produce a hard spectrum
which peaks around 0.4~\mbox{$m_{\mbox{}_\mathrm{DM}}$} (0.5~\mbox{$m_{\mbox{}_\mathrm{DM}}$}) for $\nu_{e, \mu}$
($\nu_\tau$) for the former channel and even at higher energies for
the latter~\cite{Cirelli:2005gh}. Quarks hadronize producing a large
number of mesons and baryons which subsequently decay producing a
softer neutrino spectrum. For instance, neutrinos from $b$s and $c$s
have a spectrum which peaks at energies $\sim$~0.15~\mbox{$m_{\mbox{}_\mathrm{DM}}$}~ or lower,
depending on the WIMP mass, and drops to zero around
$E_\nu \sim $~(0.6--0.7)~\mbox{$m_{\mbox{}_\mathrm{DM}}$}~\cite{Cirelli:2005gh}. Light quarks
hadronize mostly in pions which, as muons, get stopped before decaying
and therefore produce only low energy neutrinos. Once produced,
neutrinos propagate in the Sun, being absorbed via charged-current
interactions and loosing energy due to neutral-current ones. These
effects, in addition to neutrino oscillations between different
flavors, need to be taken into account~\cite{Cirelli:2005gh}.

\section{Reconstructing DM properties}

High-energy neutrinos can be detected in neutrino detectors and
stringent bounds have already been set~\cite{sklimit} and will be
further improved by neutrino telescopes~\cite{nutelescopes}. However,
as these experiments cannot reconstruct the neutrino energy, they can
only provide limited information on the neutrino spectrum. In order to
reconstruct the \emph{neutrino spectrum}, the analysis presented here
exploits the \emph{energy and angular resolution} of MIND, which allow
not only to measure precisely the energy and direction of muons
produced by the charged current interactions (CC) of $\nu_\mu$ and
$\bar{\nu}_\mu$ but possibly also the energy of the
electrons~\cite{MIND,anselmo}, produced by $\nu_e$ ($\bar{\nu}_e$)
interactions. In addition, these detectors will be able to reconstruct
the energy and angle of the hadron shower, so that a good energy and
direction resolution for the incoming neutrino will be achieved. We
show that the degeneracy among different parameters can be broken by
using the muon and electron energy spectra. In particular, a larger
annihilation rate \mbox{$\Gamma_{\mathrm{ann}} \;$} could always be traded by a smaller
branching ratio to a hard channel ($\tau^{+} \tau^{-}$) in absence of
information on the spectrum.

In the following we will consider a light WIMP candidate with $\mbox{$m_{\mbox{}_\mathrm{DM}}$} <
80$~GeV and therefore, we will focus on its dominant annihilation
modes for these masses, i.e., $\tau^{+}\tau^{-}$ (hard) and
$b\bar{b}$ (soft) channels. Note that the contribution from $c\bar{c}$
is similar to the one of $b\bar{b}$ but with a lower
normalization~\cite{Cirelli:2005gh}. This technique might not be
competitive with neutrino observatories (AMANDA and Icecube) for
WIMP masses $\gtap 100$~GeV, since the typical sizes of future MIND,
LArTPC and TASD might not allow a precise measurement of high lepton
energies. However, it should be noted that, even for high WIMP masses,
these detectors could be useful for detecting the low energy tail of
the neutrino spectrum~\cite{CMPPprep}. In our study we assume the mass
of the WIMP to be known and, for definiteness, we take $\mbox{$m_{\mbox{}_\mathrm{DM}}$} = 50$~GeV
and $70$~GeV.

Let us now comment on our assumption of taking a given value for the
WIMP mass. By the time the detectors discussed in the present article
are available, information on the WIMP mass might be already available
from the LHC (Large Hadron Collider) and, possibly, a future ILC
(International Linear Collider) as well as direct and indirect dark
matter searches. With the start of LHC in 2008, the neutral WIMP
candidate for the dark matter can be indirectly detected in an event
looking for missing energy and missing transverse momentum. By a
detailed determination of the kinematics of the quarks and leptons,
typically it might be possible to measure the WIMP mass with a 10\%
accuracy~(see, e.g., Ref.~\cite{LHCWIMP}). The prospected error
depends strongly on the model of physics beyond the SM and the values
of the parameters. For example, for  the SPS1a benchmark point, the
LSP mass could be determined to be $\mbox{$m_{\mbox{}_\mathrm{DM}}$}= (96\pm 5)$~GeV at the LHC
and $\mbox{$m_{\mbox{}_\mathrm{DM}}$}= (96\pm 0.05)$~GeV at the ILC~\cite{LHCWIMP}. For less
favorable points in the parameter space, worse accuracies will be
reachable. The value for the mass determined in collider searches
could be used as input in our analysis, assuming that the observed
WIMP corresponds to the dark matter particle.

A model-independent determination of the WIMP dark matter mass could
also be obtained in direct DM searches which look for the nuclear
recoil in an interaction between a WIMP and a nucleus in the
detector~\cite{Lewin:1995rx}. The recoil spectrum is strongly
mass-dependent if $\mbox{$m_{\mbox{}_\mathrm{DM}}$} < 100 \ \mathrm{GeV}^2$. For instance, a
superCDMS-like experiment, with exposures of $3 \times 10^3 \ (3
\times 10^4) \ [3 \times 10^5]$~kg~days and a spin-independent cross
section close to the present bound $\sigma_{\rm SI} = 10^{-7} \
\mathrm{pb}$, would be able to measure a light WIMP mass with an
accuracy of $\sim 25\% \ (15\%) \
[2.5\%]$~\cite{Green:2007rb}. Similar conclusions can be drawn for the
case of spin-dependent cross sections with comparable
statistics. Nevertheless, these measurements are affected by the
uncertainties in the WIMP velocity distribution and in the local WIMP
distribution. A method which is independent of the WIMP density and of
the WIMP-nucleon cross section has been also
proposed~\cite{Shan:2007vn}. By comparing the recoil energy spectrum
in direct searches which use different detector materials, it is found
that a WIMP mass smaller than 50~GeV can be determined with a
1-$\sigma$ error of 20\% if a total statistics of 500 events is
available~\cite{Shan:2007vn}.

Information on the WIMP mass could also be found from indirect dark
matter searches. Some information on the WIMP mass could also be
obtained from the angular distribution of neutrino-induced muons in
large water-\v{C}erenkov detectors~\cite{EG95}. On the other hand, as
the WIMP density in the center of galaxies is predicted to be
enhanced, a high rate of WIMP annihilation are expected and produce
sizable fluxes of SM particles, as neutrinos, photons, quarks and
leptons. High energy photons can be produced either directly or via
the hadronic interactions in the interstellar medium. In the first
case, the gamma spectrum presents a very clear signature with a peak
at the WIMP mass, allowing its precise determination. In the case of
gamma-rays from primary hadrons, the spectrum has a sharp energy
cut-off in correspondence with \mbox{$m_{\mbox{}_\mathrm{DM}}$}. It has been estimated that GLAST
could constrain the WIMP mass at the 25\% level, for $\mbox{$m_{\mbox{}_\mathrm{DM}}$} = 100$~GeV,
and future Imaging Atmospheric \v{C}erenkov Telescopes, with
sufficiently small energy threshold, could detect the peak from direct
annihilation into photons~\cite{DMgamma}.

In addition, the energy information in the WIMP neutrino signal in the
detectors considered in the present article could allow, in principle,
to infer directly the WIMP mass by exploiting the cutoff in the
spectra. Since the typical sizes of future MIND, LArTPC and TASD might
not allow to have sufficient statistics at the tail of the spectrum,
this measurement could be very challenging. A detailed analysis needs
to be performed, but it is beyond the scope of the present
study~\cite{CMPPprep}.

We have computed the number of electron and muon neutrino-induced
CC events for \mbox{$m_{\mbox{}_\mathrm{DM}}$} = 50~and 70~GeV, by using the evolved neutrino
spectra from Ref.~\cite{Cirelli:2005gh}. We have, conservatively,
considered nine energy bins (5 GeV bins from an energy of 5 GeV for
\mbox{$m_{\mbox{}_\mathrm{DM}}$} = 50~GeV and 7~GeV bins from an energy of 7~GeV for \mbox{$m_{\mbox{}_\mathrm{DM}}$} =
70~GeV), where the low energy cut is adopted to avoid the atmospheric
neutrino background. With good statistics, the precision in the
measurement of \mbox{$m_{\mbox{}_\mathrm{DM}}$}, for $\mbox{$m_{\mbox{}_\mathrm{DM}}$} \gtap 50$~GeV, depends on the muon
energy resolution of MIND at energies $E_\nu\sim50-100$~GeV, expected
to be $\sim 5-8\%$ at $50$~GeV and slightly worse for higher
energies~\cite{anselmo}.

For an ideal detector with perfect lepton angular resolution, the
atmospheric neutrino background would be reduced to the fraction of
events within the angular size of the Sun ($\sim 6.7\times
10^{-5}$~sr). However, the lepton does not travel in the same
direction as the incident neutrino except when averaged over many
events. We conservatively take the root mean square (rms) spread in
direction between the incident neutrino and the lepton (in radians) as
\begin{equation}
\theta_{\rm rms}\simeq \sqrt{\frac{1\ \textrm{GeV}}{E_\nu}}~,
\label{eq:rms}
\end{equation}
where $E_\nu$ is the energy of the incoming neutrino. Therefore, a
very conservative approach is to sum over the atmospheric neutrino
background contributions within a region of opening angle
$\theta_{\rm rms}$.

The expected number of muon (electron) neutrino-induced events in the
$i$-th bin is computed as a function of the WIMP-proton scattering
cross section and of the branching ratios into $b^{+}b^{-}$
(soft channel) and $\tau^{+}\tau^{-}$ (hard channel),
\begin{eqnarray}
N_{i,\mu(e)} & = & N_{\rm T} \, t \, \int_{E_i}^{E_i
  +\Delta}dE_\nu \int d\theta \, \left(\phi_{\nu_\mu(\nu_e)} (E_\nu) \,
\sigma^{\rm CC}_{\nu_\mu(\nu_e)} (E_\nu) \right. \nonumber \\
 & & \left. + \, \phi_{\bar{\nu}_\mu(\bar{\nu}_e)} (E_\nu) \,
\sigma^{\rm CC}_{\bar{\nu}_\mu(\bar{\nu}_e)} (E_\nu) \right) \times
\frac{V_{\mu(e)}}{V_{\rm det}}~,
\label{eq:events}
\end{eqnarray}
where $\Delta$ is the energy bin width, $\theta$ is the angle which
measures the relative position of the Sun with respect to the detector
location, $N_{\rm T}$ is the number of available targets, $V_{\rm
  det}$ is the total volume of the detector, $t$ is the exposure time,
$\phi$ is the evolved (anti)neutrino spectra, $\sigma^{\rm CC}$ is the
CC (anti)neutrino cross section and $V_{\mu, e}$  is the volume of
detector available for the neutrino to interact. For electron-like
events, which are all contained, this fiducial volume $V_e$ is simply
the detector volume $V_{\rm det}$. For muon-like contained-events, it
depends on the detector geometry and on the muon range in iron $R_\mu
(E_\mu)$. For a detector with cylindrical shape, it is given
by~\cite{AS01}
\begin{eqnarray}
V_\mu(E_\mu,\theta) & = & 2 h r^2
\arcsin\left(\sqrt{1-\frac{R^2_\mu(E_\mu)}{4 \, r^2}\sin^2 \theta}
\right) \nonumber \\
& & \times \left(1-\frac{R_\mu(E_\mu)}{h}|\cos\theta|\right)~,
\label{eq:range}
\end{eqnarray}
where $r=13$~m and $h=20$~m are the detector radius and height we have
considered. We approximate the muon energy by its averaged value,
which for the energy range of interest is $\langle E_\mu \rangle =
0.48 \, E_\nu$~\cite{GQRS95}. Also notice that the volume of detector
available is a function of the azimuthal angle $\theta$, which varies
from 0 to $\pi$ in 12 hours. Thus, for the assumed geometry, the
effective volume will be maximal when the Sun is at the horizon.  On
the other hand, if a good energy resolution is achieved up to some
energies even for partially-contained events (by means of using the
bending of the muon trajectory due to the magnetic field), the
fiducial volume to be considered is $V_{\rm det}$. We will consider
both cases, when partially-contained muon-like events are not included
in the analysis and when all muon-like events with the interaction
vertex in the detector contribute.

The expected number of muon and electron neutrino-induced events is
then fitted performing a $\chi^2$ analysis. We define
\begin{equation}
\chi_{\mu^{\pm} + e^{\pm}}^2 = \sum_{\alpha, \alpha'= \mu,
  e}\sum_{i,j} (n_{i,\alpha} - N_{i, \alpha})
C_{i,\alpha:,j,\alpha'}^{-1} (n_{j,\alpha'} - N_{j,\alpha'})\,,
\end{equation}
where $C$ is the covariance matrix and only statistical errors, being
the dominant ones, have been considered. $N_{i,\alpha}$ is given
by Eq.~(\ref{eq:events}) and $n_{i,\alpha}$ represents the simulated
muon (electron) data taking into account the total muon or electron
atmospheric neutrino background within the $i$-th energy bin and
integrated over a solid angle $\Delta \Omega= 2\pi \theta_{\rm rms}$,
and is given by
\begin{equation}
n_{i,\alpha}=\textrm{Smear}(N_{i,\alpha} + N_{i,{\rm atmos}})
-N_{i,{\rm atmos}}~,
\end{equation}
where the function Smear indicates that Gaussian or Poisson smearing
has been applied following the Monte Carlo techniques of
Ref.~\cite{PDG} to mimic the statistical uncertainty.

In Figs.~\ref{fig:fig1} and \ref{fig:fig2}, we illustrate the
simultaneous extraction of the scattering cross section of the WIMP
off a proton and the branching ratio into $\tau^{+} \tau^{-}$ assuming
a $50$~kton MIND with the geometry described above. In
Fig.~\ref{fig:fig1} we consider the case of a WIMP mass of \mbox{$m_{\mbox{}_\mathrm{DM}}$} =
50~GeV and the input simulated values are $\sigma = 7 \times 10^{-3}$
pb and BR$_{\tau^+\tau^-}$ = $20\%$. We have assumed 15 years of data
taking and have only included electron-like and contained muon-like
events. The dotted-dashed red lines depict the 90\%~confidence level
(CL) contours assuming that no energy information is available: as
previously anticipated, a larger annihilation rate could always be
mimicked by a lower branching ratio into $\tau^{+} \tau^{-}$. Notice
that there will be a continuous region of degenerate solutions,
implying the impossibility of extracting either the different
branching ratios or the WIMP-proton scattering cross section. In this
scenario, the measurement of the WIMP mass is of course out of
reach. Instead, if we now include the information on the neutrino
energy spectrum, we obtain the $90\%$~CL contours depicted by the
solid blue line. The strong correlation between $\sigma$ and
BR$_{\tau^+\tau^-}$ (or BR$_{b^+b^-}$) is broken. The reason is
simple: the differential neutrino spectrum for each possible
annihilation channel (hard or soft) has a characteristic shape,
different from one channel to another. For this case, \mbox{$m_{\mbox{}_\mathrm{DM}}$} = 50~GeV,
the spectrum for $b\bar{b}$ peaks at $\sim 7$~GeV and represents
roughly $20\%$ of that of $\tau^{+}\tau^{-}$, peaked at $\sim
20$~GeV. We also illustrate the results for the ideal case for which
the atmospheric neutrino background only contributes within the real
angular size of the Sun (dashed magenta lines). As can be seen, by
considering a more detailed and realistic expression for the angular
resolution than that given by Eq.~(\ref{eq:rms}), our results would
not be significantly improved. In Fig.~\ref{fig:fig2} we show the
90\%~CL contours for 15 years for $\mbox{$m_{\mbox{}_\mathrm{DM}}$}=70$~GeV with the input
simulated values $\sigma= 5 \times 10^{-3}$~pb and BR$_{\tau^+\tau^-}
= 10\%$. The solid blue line represents the case when
partially-contained muon-like events are not considered, whereas the
inner dotted magenta line depicts the case when we also include this
type of events to perform the analysis. In this figure, we have
considered the atmospheric neutrino background integrated over a
half-cone aperture given by Eq.~(\ref{eq:rms}). It is important to
note that by adding the partially-contained events, in order to
achieve the same results as for the case of only contained events,
the total number of years could be reduced from 15 to 10 years of data
taking, the minimum running time of these detectors in the context of
neutrino factories. In both figures, the equivalent $90\%$ CL SK
limit~\cite{sklimit} is included.

\begin{figure}[t]
\begin{center}
\includegraphics[width=3.5in]{mchi50_v7.ps}
\caption{\footnotesize
The solid blue (dotted-dashed red) contours denote the $90\%$ CL
limits, for 2 degrees of freedom, for the simultaneous extraction of
the WIMP-proton cross section and the annihilation branching ratio
into the $\tau^{+} \tau^{-}$ channel with (without) energy
information by using only fully-contained events with an exposure of
750~kton $\cdot$ yrs (see text for details). The simulated nature
values are $\mbox{$m_{\mbox{}_\mathrm{DM}}$}=50$~GeV, $\sigma = 7 \times 10^{-3}$~pb and ${\rm
  BR}_{\tau^+\tau^-}=20\%$. The dashed magenta lines (inner contours)
illustrate the case (for 90\%~CL) in which the atmospheric neutrino
background is integrated only over the real size of the Sun. We also
show the 90\%~CL excluded region by SK~\cite{sklimit}.
}
\label{fig:fig1}
\vspace{-2mm}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics[width=3.5in]{mchi70_v6.ps}
\caption{\footnotesize The solid blue (dotted-dashed red) contours
  denote the $90\%$ CL limits, for 2 degrees of freedom, for the
  simultaneous extraction of the WIMP-proton cross section and the
  annihilation branching ratio into the $\tau^{+} \tau^{-}$ channel
  with (without) energy information by using only fully-contained
  events with an exposure of 750~kton $\cdot$ yrs (see text for
  details). The simulated nature values are $\mbox{$m_{\mbox{}_\mathrm{DM}}$} =70$~GeV, $\sigma =
  5 \times 10^{-3}$~pb and ${\rm BR}_{\tau^+\tau^-}=10\%$. The dashed
  magenta lines (inner contours) illustrate the case (for 90\%~CL) in
  which partially-contained events are also included in the
  analysis. We also show the 90\%~CL excluded region by
  SK~\cite{sklimit}.
}
\label{fig:fig2}
\end{center}
\end{figure}

As both figures show, for these relatively low WIMP masses and large
spin-dependent cross sections, the determination of the annihilation
branching ratios and WIMP-proton cross section could be
achieved. Notice that a positive signal of neutrinos from WIMP
annihilations in the Sun in these detectors will immediately point to
a spin-dependent cross section. In LArTPC and TASD similar results
could be obtained. In fact, we have checked that considering only the
$\nu_\mu$ or the $\nu_e$ signal does not modify the results
substantially, a part from a small change due to the smaller
atmospheric neutrino background for $\nu_e$. A detailed analysis for
different WIMP masses and type of detectors is in
preparation~\cite{CMPPprep}.

\section{Conclusions}

Searches of high-energy neutrinos from WIMP annihilations in the Sun
could constitute a powerful probe of WIMP properties. While the total
neutrino flux is controlled by the annihilation rate (proportional to
the WIMP-proton cross section), the shape of the neutrino spectrum
depends on the WIMP mass and on the branching ratios of different
channels. \v{C}erenkov neutrino detectors/telescopes are counting
experiments and can only provide limited information. On the contrary,
we have shown that, by exploiting the good energy and angular
resolution at energies of tens of GeV, future MIND, LArTPC and TASD
could have the capability to reconstruct the neutrino spectrum, and
could provide important information on the WIMP mass, its annihilation
branching ratios and WIMP-proton cross section. In particular, our
analysis shows that, for a given value of the WIMP mass, the
degeneracy between the WIMP-proton cross section and the branching
ratios into soft and hard channels could be broken and important
information on the WIMP dark matter couplings might be obtained. This
information should be combined with the constraints and measurements
from direct and indirect DM searches to probe the nature and
properties of DM particles and to test that the WIMP candidate found
in collider searches does indeed constitute the observed dark matter
of the Universe.

\section*{Acknowledgments}

It is pleasure to thank A.~Cervera, P.~Lipari and G.~Weiglein for
useful discussions. OM is supported by the European Programme ``The
Quest for Unification'', contract MRTN-CT-2004-503369.  SPR is
partially supported by the Spanish Grant FPA2005-01678 of the MCT. SP
is partially supported by CARE, contract number
RII3-CT-2003-506395. OM and SP would like to thank the Theoretical
Physics Department at Fermilab for hospitality.