text:
\section{Introduction: Why Inflation Works}
\label{sec:introduction}

Inflationary cosmology \cite{Guth:1980zm,Linde:1981mu,Albrecht:1982wi} is the most successful and widely accepted paradigm for understanding the very
early universe. By all measures inflation is a compelling and scientifically useful theory, and makes quantitative predictions which have been strongly
supported by recent data \cite{Dunkley:2008ie,Komatsu:2008hk,Kinney:2008wy}. The two main hallmarks of inflationary cosmology are solutions to the
flatness and horizon problems of the standard Big Bang cosmology: why is the universe so close to geometrically flat, and how did the apparent acausal
structure of the universe arise? ``Acausal'' more specifically means that the universe is approximately homogeneous on scales larger than a Hubble
length $H^{-1}$, and in addition exhibits a spectrum of density perturbations which is correlated on scales larger than a Hubble length.
Such {\it superhorizon} correlations are generated in inflation by accelerated expansion, $\ddot a / a > 0$, where $a\left(t\right)$ is the
cosmological scale factor, which means that the comoving Hubble length $d_H \simeq (a H)^{-1}$ shrinks with the expansion of the universe,
\begin{equation}
\frac{d}{d \ln{a}} \left(a H\right)^{-1} < 0.
\end{equation}
Therefore, quantum perturbations, which have constant wavelength in comoving units, are smaller than the Hubble length at early times, and are
redshifted to larger than the Hubble length at late times, where they are ``frozen'' as classical perturbations. Furthermore, as long as the Hubble
constant $H$ is slowly varying with time, the perturbations generated in inflation are nearly scale-invariant, consistent with observation.
Furthermore, the solution to the horizon problem and the flatness problem are linked in inflation via a conservation law,
\begin{equation}
\label{eq:conservationlaw}
\frac{d}{d \ln{a}} \frac{\left\vert\Omega - 1\right\vert}{d_H^2} = 0.
\end{equation}
Through this conservation law, a universe with shrinking comoving horizon size is identical to a universe which is evolving toward flatness,
\begin{equation}
\frac{d}{d \ln{a}} \left\vert\Omega - 1\right\vert < 0.
\end{equation}
Inflation therefore solves the horizon and flatness problems of the standard Big Bang with a {\it single} mechanism: accelerated expansion.

However, inflation is not the only way to accomplish this goal, as
can be seen from the fact that the acceleration $\ddot a$ appears
nowhere in the conservation law (\ref{eq:conservationlaw}). To
solve both the horizon and flatness problems, it is sufficient to
have a shrinking comoving Hubble length. One way to do this is
accelerated expansion, but another is to have a collapsing
universe, $H = \left(\dot a / a\right) < 0$. A collapsing, matter-
or radiation-dominated universe also has a shrinking comoving
Hubble length, which will generate perturbations in a manner
similar to inflation. This is the mechanism used by the Ekpyrotic
scenario \cite{Gratton:2003pe} to construct a cosmology consistent
with observations. It is also possible to decouple the horizon and
flatness problems, for example in theories with a varying speed of
light, so that the causal horizon is much larger than the Hubble
length \cite{Albrecht:1998ir}. Such theories can in principle
solve the horizon problem, but not the flatness problem, since the
conservation law (\ref{eq:conservationlaw}) is violated. It is
also possible to solve the horizon problem by a universe which is
much older than a Hubble time as in string gas cosmology
\cite{Brandenberger:2009jq} or island cosmology \cite{Dutta:2005gt,Dutta:2005if}, or by the inclusion of extra
dimensions \cite{Starkman:2000dy,Starkman:2001xu}. However, it has been argued that
inflation and ekpyrosis are the {\it only} mechanisms for
generating a scale-invariant spectrum of perturbations
\cite{Gratton:2003pe,Khoury:2008wj}.

In this paper, we discuss a method of solving the cosmological horizon problem and seeding scale-invariant primordial perturbations in a cosmology with decelerating expansion and a corresponding {\it growing} comoving Hubble horizon. The key to implementing such a model is the fact that curvature perturbations are not generated at the Hubble horizon, but at the acoustic horizon determined by the speed of sound of a scalar field. For canonical field theories, the two are identical, but for non-canonical field theories, they are not. If one has a decaying, superluminal sound speed, curvature perturbations can be generated outside the Hubble horizon without inflation.  We propose the term
 {\it tachyacoustic} for such cosmologies, which are closely related to varying speed of light theories. This idea has some history: such cosmologies were first proposed by Armendariz-Picon in the context of modified dispersion relations \cite{ArmendarizPicon:2006if}, and the generation of perturbations in such cosmologies was further considered by Piao \cite{Piao:2006ja}. The idea re-emerged in the context of varying speed of light theories by Magueijo \cite{Magueijo:2008pm}, and non-canonical Lagrangians by Magueijo \cite{Magueijo:2008sx} and Piao \cite{Piao:2008ip}. In this paper, we outline a general approach to such cosmologies based on the generalization of the inflationary flow formalism \cite{Kinney:2002qn} introduced by Bean, {\it et al.} for the case of arbitrary Lagrangians \cite{Bean:2008ga}. We find that there is a class of Lagrangians with the necessary properties for tachyacoustic cosmologies, and discuss two interesting examples. We find that it is straightforward to generate nearly scale-invariant perturbations for these Lagrangians, and show that they have the property of reducing to instanton-like solutions with infinite sound speed on the initial-time boundary of the spacetime. We speculate that this property may allow a self-consistent description of tachyacoustic cosmologies within a Wheeler-deWitt description of quantum cosmology. Finally, we show that such models are causally self-consistent, and argue that they form a viable class of alternatives to inflation.

\section{Tachyacoustic Cosmology}
\label{sec:TachyacousticCosmology}

In this paper, we consider a  way of generating scale-invariant superhorizon cosmological perturbations based on non-canonical scalar field
Lagrangians with a speed of sound faster than the speed of light, $c_S > 1$.  If the universe is dominated by a scalar field with speed of sound $c_S$, the relevant horizon for
the generation of density perturbations is not the Hubble horizon $d_H \simeq (a H)^{-1}$ but the acoustic horizon,
\begin{equation}
D_H \simeq \frac{c_S}{a H}.
\end{equation}
Mode freezing at the acoustic horizon is well-known in
non-canonical inflation models, for example k-Inflation
\cite{ArmendarizPicon:1999rj} and DBI inflation
\cite{Silverstein:2003hf}. In non-canonical inflation models, the
Hubble horizon and the acoustic horizon are {\it both} shrinking
in comoving units, resulting in the generation of density
perturbations at the acoustic horizon and gravitational wave
perturbations at the Hubble horizon \cite{Garriga:1999vw}.
However, the comoving Hubble horizon need not be shrinking to
generate curvature perturbations: all that is required is that the
{\it acoustic} horizon be shrinking, $d D_H / d \ln{a} < 0$. In
this case, if curvature perturbations are to be generated on
scales larger than the Hubble horizon, it is necessary that the
acoustic horizon be larger than the Hubble horizon, which requires
a speed of sound greater than the speed of light. Such theories
were studied recently by Babichev {\it et al.}
\cite{Babichev:2006vx,Babichev:2007dw}, who showed that k-essence theories with
$c_S > 1$ are causally self-consistent (see Appendix
\ref{sec:stablecau}), and can be mapped to bimetric theories with
two ``light cones'', one given by the Hubble horizon, and the
other given by the acoustic horizon, which can be larger than the
Hubble horizon without the presence of closed timelike loops. This
opens the possibility that one can construct a decelerating
cosmology which nonetheless generates perturbations on
super-Hubble scales via a superluminal, shrinking acoustic cone.

To explicitly construct such a model, consider a Dirac-Born-Infeld (DBI) Lagrangian,
\begin{equation}
\label{eq:DBIlagrangian}
{\cal L}\left(\phi,X\right) = - f^{-1}\left(\phi\right) \sqrt{1 - f\left(\phi\right) \dot\phi^2} + f^{-1}\left(\phi\right) - V\left(\phi\right),
\end{equation}
where we take the metric to have negative signature $\eta_{\mu\nu} = {\rm diag.}(1, -1, -1, -1)$, so that the kinetic term for the field is positive,
\begin{equation}
\dot\phi^2 \equiv g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi > 0.
\end{equation}
Such Lagrangians often arise in string theory, for example in the case of DBI inflation. For the moment, we will not attempt to make a connection with
string theory, but will take the Lagrangian (\ref{eq:DBIlagrangian}) as a phenomenological {\it ansatz}. Take the background spacetime to be a flat,
Friedmann-Robertson-Walker (FRW) space, $g_{\mu \nu} = a^2 \eta_{\mu\nu}$, so that the scale factor evolves as
\begin{equation}
a \propto \exp\left(\int{H dt}\right) \propto e^{- N},
\end{equation}
where we define the number of e-folds $N$ as\footnote{We use the usual convention that $N \rightarrow \infty$ corresponds to early time, and
$N \rightarrow - \infty$ corresponds to late time.}
\begin{equation}
N \equiv - \int{H dt}.
\end{equation}
There is a class of exact solutions \cite{Chimento:2007es,Kinney:2007ag} to the equation of motion for the field $\phi$ characterized by two
dimensionless flow parameters $\epsilon$ and $s$, where
\begin{equation}
\label{eq:defofepsilon}
\epsilon \equiv \frac{1}{H} \frac{d H}{d N} = {\rm const.},
\end{equation}
and
\begin{equation}
s \equiv -\frac{1}{c_S}\frac{d c_S}{d N} = {\rm const.}
\end{equation}
The parameter $\epsilon$ has its usual interpretation in terms of the equation of state of the scalar field,
\begin{equation}
\label{eq:epsilondef} p = \rho \left(\frac{2}{3} \epsilon -
1\right).
\end{equation}
For $\epsilon = {\rm const.}$, the scale factor evolves as a power-law, $a \propto t^{1 / \epsilon}$, so that the expansion is accelerating
({\it i.e.} inflation) for $\epsilon < 1$. The speed of sound evolves as
\begin{equation}
\label{eq:DBIsoundspeed}
c_S = \sqrt{1 - f\left(\phi\right) \dot\phi^2} \propto e^{-s N},
\end{equation}
and the Hubble parameter evolves as
\begin{equation}
\label{eq:Hphi} H = \frac{\dot a}{a} \propto e^{\epsilon N}.
\end{equation}
The parameter $\epsilon$ is a positive-definite quantity for $p \geq -\rho$, so that the Hubble constant always decreases with expansion. In contrast,
the parameter $s$ can take either sign, with $s > 0$ corresponding to a sound speed which increases with expansion, and $s < 0$ corresponding to an
decreasing sound speed. (See Ref. \cite{Kinney:2007ag} for a detailed derivation of this solution.) The important dynamics for the generation of
perturbations is the time evolution of the corresponding horizons in comoving units. The comoving Hubble horizon evolves as
\begin{equation}
d_H \propto \left(a H\right)^{-1} \propto e^{\left(1 - \epsilon\right) N} \propto \tau,
\end{equation}
where $\tau$ is the conformal time. The Hubble horizon is shrinking in comoving units for $\epsilon < 1$, which is identical to accelerated expansion,
and is the usual condition for inflation. The acoustic horizon behaves as
\begin{equation}
D_H \propto \frac{c_S}{a H} \propto e^{\left(1 - \epsilon - s\right) N} \propto \tau^{\left(1 - \epsilon - s\right) / \left(1 - \epsilon\right)}.
\end{equation}
Therefore the condition for a shrinking acoustic horizon, $1 -
\epsilon - s > 0$, is {\it not} identical to accelerated
expansion. For $\epsilon > 1$ and $s < 1 - \epsilon$, the
expansion is non-inflationary, the Hubble horizon is growing in
comoving units, and the acoustic horizon is shrinking. The initial
singularity is at $\tau = 0$, and we see immediately that for the
tachyacoustic solution, the speed of sound in the scalar field is
{\it infinite} at the initial singularity, and the acoustic
horizon is likewise infinite in size. Therefore, such a cosmology
presents no ``horizon problem'' in the usual sense, since even a
spatially infinite spacetime is causally connected on the
initial-time boundary. Furthermore, unlike in the case of
inflation, there is no period of reheating necessary, since the
cosmological evolution can be radiation-dominated throughout and
the cosmic temperature is not driven exponentially to zero.

In the next section, we use the the generalized flow function approach of Bean, {\it et al.} \cite{Bean:2008ga} to construct a class of Lagrangians
with solutions of the type outlined above, with constant flow parameters. In these solutions, the scale-factor evolves as a power-law in time and the
equation of motion for curvature perturbations can be solved exactly, which we discuss in Sec. \ref{sec:Perturbations}.

\section{Flow hierarchy for general k-essence models}
\label{sec:flowhier}

We now generalize the discussion in the last section to an arbitrary k-essence model. Consider a general Lagrangian of the form
${\cal{L}}={\cal{L}}\left[X,\phi\right]$, where $2X=g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ is the canonical kinetic term ($X>0$ according
to our choice of the metric signature). The energy density $\rho$ and pressure $p$ are given by
\begin{eqnarray}
p &=& {\cal L}\left(X,\phi\right),\\
\rho &=& 2 X {\cal L}_X - {\cal L}.
\end{eqnarray}
The speed of sound is given by
\begin{eqnarray}
\label{defspeedofsound} c_S^{2} &\equiv& \frac{p_X}{\rho_X} = \left(1 + 2X\frac{{\cal
L}_{XX}}{{\cal L}_{X}}\right)^{-1},
\end{eqnarray}
where the subscript ``${X}$" indicates a derivative with respect to the kinetic
term.  Throughout this section, unless otherwise stated, we will follow closely Bean {\it{et al.}} \cite{Bean:2008ga}. We define the first three
{\it flow parameters} as derivatives with respect to the number of e-folds, $dN = - H dt$:\footnote{The parameters $s$ and $\tilde s$ correspond to
the parameters $\kappa$ and $\tilde\kappa$ in Bean, {\it et al.} \cite{Bean:2008ga}}.
\begin{eqnarray}
\label{epsN}
\epsilon &\equiv& \frac{1}{H} \frac{d H}{dN}
,\\ \label{sN}
s &\equiv& -\frac{1}{c_S} \frac{d c_S}{dN},\\ \label{stilN}
\tilde{s} &\equiv& \frac{1}{{\cal{L}}_{X}}\frac{d {\cal{L}}_{X}}{dN}.
\end{eqnarray}
The Friedmann equation can be written in terms of the reduced Planck mass  $M_P=1/\sqrt{8\pi G}$
\begin{equation}
H^2 = \frac{1}{3 M_P^2} \rho = \frac{1}{3 M_P^2} \left(2X {\cal
L}_X - {\cal L}\right),
\end{equation}
and the continuity equation is
\begin{equation}
\dot\rho = 2 H {\dot H} = -3 H \left(\rho + p\right) = - 6 H X {\cal L}_X.
\end{equation}
For monotonic field evolution, the field value $\phi$ can be used as a ``clock'', and all other quantities expressed as functions of $\phi$, for
example $X = X\left(\phi\right)$, ${\cal L} = {\cal L}\left[X\left(\phi\right),\phi\right]$, and so on. We consider the homogeneous case, so that
$\dot\phi = \sqrt{2 X}$. Using
\begin{equation}
\frac{d}{dt} = \dot\phi \frac{d}{d \phi} = \sqrt{2 X} \frac{d}{d\phi},
\end{equation}
we can re-write the Friedmann and continuity equations as the {\it Hamilton Jacobi} equations,
\begin{eqnarray}
\label{hamjac1} \dot
\phi = \sqrt{2 X} &=& -\frac{2M_P^2}{{\cal L}_{X}}H'(\phi),\\ \label{hamjac2}
3M_P^2H^2(\phi)&=&\frac{4M_P^4{H'\left(\phi\right)}^2}{{\cal L}_{X}}-{\cal L}.
\end{eqnarray}
where a prime denotes a derivative with respect to the field $\phi$. The number of e-folds $dN$ can similarly be re-written in terms of $d\phi$ by:
\begin{eqnarray}
\label{eq:Nphi}
dN \equiv -H dt &&= - \frac{H}{\sqrt{2 X}} d\phi\\
&&= \frac{{\cal L}_X}{2 M_P^2} \left(\frac{H\left(\phi\right)}{H'\left(\phi\right)}\right) d\phi.
\end{eqnarray}
The flow parameters $\epsilon$, $s$, and $\tilde s$ (\ref{epsN}) can therefore be written as derivatives with respect to the field $\phi$ as:
\begin{eqnarray}
\label{defeps}
\epsilon\left(\phi\right) &=& \frac{2
M_P^2}{{\cal{L}}_{X}}
\left(\frac{H'\left(\phi\right)}{H\left(\phi\right)}\right)^2,\\ \label{defs}
s\left(\phi\right) &=& - \frac{2
M_P^2}{{\cal{L}}_{X}}
\frac{H'\left(\phi\right)}{H\left(\phi\right)} \frac{c_S'\left(\phi\right)}{c_S\left(\phi\right)},\\ \label{defstil}
\tilde{s}\left(\phi\right) &=& \frac{2
M_P^2}{{\cal{L}}_{X}}
\frac{H'\left(\phi\right)}{H\left(\phi\right)}\frac{{\cal{L'}}_{X}}{{\cal{L}}_{X}}.
\end{eqnarray}
Taking successive derivatives $d/dN$ with respect to the number of e-folds yields an infinite hierarchy of flow equations
\cite{Kinney:2002qn,Bean:2008ga},
\begin{eqnarray}
\label{flowequations} \frac{d \epsilon}{d N} &=&
-\epsilon\left(2 \epsilon - 2 \tilde{\eta} + \tilde{s}\right),\cr
\frac{d \tilde{\eta}}{d
N} &=& -\tilde{\eta}\left(\epsilon + \tilde{s}\right) + {}^2 \lambda,\cr
\frac{d s}{d N} &=& -s \left(\epsilon -
\tilde{\eta}+\tilde{s}+s\right) + \epsilon \rho,\cr
\cr\frac{d \tilde{s}}{d N} &=& -\tilde{s} \left(\epsilon -
\tilde{\eta}+2 \tilde{s} \right) + \epsilon {}^1 \beta,
\cr \frac{d
{}^\ell \lambda}{d N} &=& -{}^\ell \lambda \left[\ell \left(\tilde{s} +
\epsilon\right) - \left(\ell - 1\right) \tilde{\eta}\right] + {}^{\ell +
1}\lambda,
\cr \frac{d {}^\ell \alpha}{d N} &=& -{}^\ell \alpha
\left[\left(\ell - 1\right)
(\epsilon-\tilde{\eta})+\ell\tilde{s} + s\right] + {}^{\ell + 1} \alpha,\cr
\frac{d {}^\ell \beta}{d N} &=& -{}^\ell \beta
\left[\left(\ell - 1\right)
(\epsilon-\tilde{\eta})+\left(\ell+1\right)\tilde{s}\right] + {}^{\ell + 1} \beta,
\end{eqnarray}
where the higher-order flow parameters are defined as follows, where $\ell = 1,\ldots\infty$ is an integer parameter:
\begin{eqnarray}
\label{hflowpar}
\tilde{\eta}\left(\phi\right) &=& {}^1 \lambda = \frac{2
M_P^2}{{\cal{L}}_{X}}
\frac{H''\left(\phi\right)}{H\left(\phi\right)},\cr
{}^\ell \lambda\left(\phi\right) &=& \left(\frac{2
M_P^2}{{\cal{L}}_{X}}\right)^{\ell}
\left(\frac{H'\left(\phi\right)}{H\left(\phi\right)}\right)^{\ell
- 1} \frac{1}{H\left(\phi\right)} \frac{d^{\ell + 1}
}{d \phi^{\ell + 1}}H\left(\phi\right),\cr
{}^\ell
\alpha\left(\phi\right) &=& \left(\frac{2
M_P^2}{{\cal{L}}_{X}}\right)^{\ell}
\left(\frac{H'\left(\phi\right)}{H\left(\phi\right)}\right)^{\ell
- 1} \frac{1}{c_S^{-1}(\phi)} \frac{d^{\ell + 1}
}{d \phi^{\ell + 1}}c_S^{-1}(\phi),
\cr {}^\ell
\beta\left(\phi\right) &=& \left(\frac{2
M_P^2}{{\cal{L}}_{X}}\right)^{\ell}
\left(\frac{H'\left(\phi\right)}{H\left(\phi\right)}\right)^{\ell
- 1} \frac{1}{{\cal{L}}_{X}} \frac{d^{\ell + 1}
}{d \phi^{\ell + 1}}{\cal{L}}_{X}.
\end{eqnarray}
Solutions to this infinite hierarchy of flow equations are equivalent to solutions of the scalar field equation of motion. In the next section, we
specialize to the case where the  flow parameters are constant, which results in an exactly solvable system.

\section{Cosmological solutions for constant flow parameters}
\label{sec:flowsolutions}

The simplest way to solve the flow equations derived in the preceding section is to take all of the flow parameters to be constant,
\begin{equation}
\frac{d \epsilon}{d N} = \frac{d s}{d N} = \frac{d \tilde s}{d N} = \frac{d {}^\ell\lambda}{dN} = \frac{d {}^\ell\alpha}{dN}  = \frac{d {}^\ell\beta}{dN} = 0.
\end{equation}
Then, from (\ref{epsN}-\ref{stilN}) we easily find the following relations:
\begin{eqnarray}
\label{solflowparN}
H &\propto& e^{\epsilon N}
,\cr
c_S  &\propto&  e^{-s N},\cr
{\cal{L}}_{X} &\propto& e^{\tilde{s}N},
\end{eqnarray}
The first two are identical to the DBI case, Eqs.
(\ref{eq:DBIsoundspeed}) and (\ref{eq:Hphi}), but in the fully
general case ${\cal L}_X$ evolves independently of $c_S$. It is
straightforward to verify that the full flow hierarchy
(\ref{hflowpar}) reduces to an exactly solvable set of algebraic
equations, with the higher-order parameters expressed in terms of
$\epsilon$, $s$, and $\tilde s$. We can use the relations
(\ref{defeps},\ref{defs},\ref{defstil}) to solve for
$H\left(\phi\right)$, $c_S\left(\phi\right)$, and ${\cal
L}_X\left(\phi\right)$ as follows: from Eqs.
(\ref{defeps},\ref{defstil}), we have
\begin{equation}
{\tilde s} = \frac{2 M_P^2}{{\cal L}_X} \left(\frac{H'}{H}\right)\frac{{\cal L}_X'}{{\cal L}_X} = M_p \sqrt{2 \epsilon} \frac{{\cal L}_X'}{{\cal L}_X} = {\rm const.}
\end{equation}
We then have a differential equation for ${\cal L}_X$,
\begin{equation}
\frac{{\cal{L'}}_{X}}{{\cal{L}}_{X}^{3/2}}=\frac{\tilde{s}}{
M_P \sqrt{2 \epsilon}} = {\rm const.},
\end{equation}
with solution
\begin{equation}
\label{lagXphi}
{\cal{L}}_{X}\left(\phi\right)=\frac{8\epsilon}{\tilde{s}^2}\left(\frac{M_P}{\phi}\right)^{2},
\end{equation}
where the integration constant has been absorbed into a field
redefinition. From Eq. (\ref{solflowparN}), the field then evolves
as
\begin{equation}
\phi^2 \propto e^{-{\tilde s} N},
\end{equation}
so that the direction of the field evolution depends on the sign of $\tilde s$,
\begin{equation}
\frac{d \phi}{\phi} = -\frac{\tilde s}{2} d N.
\end{equation}
Equation (\ref{defeps}) then reduces to
\begin{equation}
\left(\frac{H'}{H}\right)^2 = \frac{4 \epsilon^2}{{\tilde s}^2 \phi^2},
\end{equation}
with solution
\begin{equation}
H \propto \phi^{\pm 2 \epsilon / \tilde s}.
\end{equation}
The sign ambiguity can be resolved by requiring that the universe be expanding, $d H / d N > 0$, so that
\begin{equation}
H \propto \phi^{-2 \epsilon / {\tilde s}} \propto e^{\epsilon N}.
\end{equation}
Finally, we solve for the speed of sound using Eq. (\ref{defs}), which reduces to
\begin{equation}
\frac{c_S'}{c_S} = \frac{2 s}{\tilde s} = {\rm const.},
\end{equation}
with solution
\begin{equation}
c_S \propto \phi^{2 s / \tilde s}.
\end{equation}
Since our choice of $N = 0$ corresponds to an arbitrary renormalization of the scale factor $a \propto e^{-N}$, we can without loss of generality
define $c_S = 1$ at $N = 0$, so that the general solution for the background evolution is given by
\begin{equation}
\label{sqrtlagX}
{\cal{L}}_{X} = \frac{8\epsilon}{{\tilde s}^2}\left(\frac{M_P}{\phi}\right)^2,
\end{equation}
\begin{equation}
\label{hubpar}
H\left(\phi\right)=H_0\left(\frac{\phi}{\phi_0}\right)^{-2\epsilon/\tilde{s}},
\end{equation}
\begin{equation}
\label{gamm}
c_S\left(\phi\right)=\left(\frac{\phi}{\phi_0}\right)^{2s/\tilde{s}},
\end{equation}
where the field evolves as
\begin{equation}
\label{efoldsphi}
\frac{\phi}{\phi_0} = e^{-{\tilde s} N / 2}.
\end{equation}

We can derive the time dependence of the scale factor using the Hamilton-Jacobi equation (\ref{hamjac1}),
\begin{eqnarray}
\label{dotphi} \dot
\phi&=&\frac{\tilde{s}}{2} H\left(\phi\right) \phi = \sqrt{2 X},
\end{eqnarray}
so that the kinetic term can be written as
\begin{eqnarray}
\label{Xphi}
X\left(\phi\right)&=\frac{\tilde{s}^2}{8} H^2\left(\phi\right) \phi^2 .
\end{eqnarray}
Integrating expression (\ref{dotphi}) gives
\begin{equation}
\label{hubpart}
H\left(t\right)=\frac{1}{\epsilon t},
\end{equation}
so that the scale factor evolves as a power-law in time, consistent with the relation (\ref{eq:defofepsilon}) between $\epsilon$ and the equation of
state $w = p / \rho$,
\begin{equation}
\label{scalefac}
a\left(t\right)\propto t^{1/\epsilon} = t^{2 / 3 \left(1 + w\right)}.
\end{equation}
Radiation-dominated evolution therefore corresponds to $\epsilon = 2$, and matter-dominated evolution corresponds to $\epsilon = 3/2$. Inflation
corresponds to $\epsilon < 1$. The comoving Hubble horizon evolves proportional the to the conformal time,
\begin{equation}
d_H \propto (a H)^{-1} \propto e^{\left(1 - \epsilon\right) N} \propto \tau,
\end{equation}
and the acoustic horizon evolves as
\begin{equation}
D_H \propto \frac{c_S}{a H} \propto e^{\left(1 - \epsilon - s\right) N} \propto \tau^{\left(1 - \epsilon - s\right) / \left(1 - \epsilon\right)},
\end{equation}
identically to the DBI case discussed in Sec. \ref{sec:TachyacousticCosmology}. For $\epsilon > 1$, the acoustic horizon is shrinking in comoving units
for $s < 1 - \epsilon$. Note that this behavior is independent of the parameter $\tilde s$, which determines the form of the Lagrangian, as we discuss
in the next section.

\section{Reconstructing the action}
\label{sec:reconstruction}

In the past two sections we have solved the flow hierarchy for a
model characterized by constant flow parameters, which allowed us
to solve for $H\left(\phi\right)$, $c_S\left(\phi\right)$, and
${\cal L}_X\left(\phi\right)$; only the derivative of the
Lagrangian with respect to the kinetic term $X$ is determined.
Therefore this solution corresponds not to a single action but a
class of actions. In this section we derive a general equation for
Lagrangians in this class, and discuss two specific examples.

From Eqs. (\ref{lagXphi}) and (\ref{gamm}), we see that the speed of sound $c_S$ can be written in terms of ${\cal{L}}_{X}$
\begin{equation}
\label{lagXgamma}
c_S^2=C^{-1} {\cal{L}}_{X}^{-2 s/\tilde{s}} = \left[1 + 2 X \frac{{\cal L}_{XX}}{{\cal L}_X}\right]^{-1},
\end{equation}
where we have used Eq. (\ref{defspeedofsound}), and defined
\begin{equation}
\label{defK}
C \equiv \left(\frac{\tilde{s}^2\phi_0^2}{8M_P^2\epsilon}\right)^{2 s / \tilde s}.
\end{equation}
The result is a differential equation for the function
${\cal{L}}\left(X,\phi\right)$:
\begin{equation}
\label{difflag}
2X{\cal{L}}_{XX}+{\cal{L}}_X-C {\cal{L}}_{X}^{n}=0,
\end{equation}
where we have defined
\begin{equation}
\label{difflagconst}
n\equiv 1 + \frac{2 s}{\tilde{s}}.
\end{equation}
Therefore, by specifying a relationship between the parameters $s$ and $\tilde s$, we can construct a Lagrangian as the solution to the differential
equation (\ref{difflag}). For example, a canonical Lagrangian with speed of sound $c_S = {\rm const.} = 1$ is just the case $s = 0$, so that $n = 1$
and $C = 1$, and Eq. (\ref{difflag}) becomes
\begin{equation}
{\cal{L}}_{XX}=0,
\end{equation}
with general solution
\begin{equation}
{\cal L} = f\left(\phi\right) X - V\left(\phi\right).
\end{equation}
Here $f\left(\phi\right)$ and $V\left(\phi\right)$ are free functions which arise from integration of the second-order equation (\ref{difflag}).
The function $f\left(\phi\right)$ can be eliminated by a field redefinition $d \varphi = \sqrt{f\left(\phi\right)} d \phi$, resulting in a manifestly
canonical Lagrangian for $\varphi$, as we would expect from setting $c_S = 1$. A canonical Lagrangian can support inflationary solutions, but not
tachyacoustic solutions, and is therefore not of interest here. However, other choices of $n$ do yield tachyacoustic solutions, and we focus on two
such choices:
\begin{enumerate}
\item{$n=0$: A Cuscuton-like model.}
\item{$n=3$: A DBI model.}
\end{enumerate}
We discuss each case separately below.

\subsection{{$\mathbf{n=0}$\bf{: A Cuscuton-like model}}}
\label{subsec:cuscutonlike}

The case $n=0$ corresponds to $\tilde{s}=-2s$ in (\ref{difflagconst}), with solution
\begin{equation}
\label{lagcuscs}
{\cal{L}}\left(X,\phi\right)=2f\left(\phi\right)\sqrt{X} + C X - V\left(\phi\right).
\end{equation}
This Lagrangian is similar to a ``cuscuton'' Lagrangian \cite{Afshordi:2006ad}, with the addition of a term proportional to $X$.  Unlike the original
Cuscuton model, which represents a causal field
with infinite speed of sound, the solution obtained here is valid for the general case, in which the speed of sound can be finite. A similar cuscuton-like Lagrangian was considered in Ref. \cite{Piao:2008ip}.

As in the canonical case, the functions $f\left(\phi\right)$ and $V\left(\phi\right)$ are free functions resulting from integrating Eq. (\ref{difflag}). Unlike the canonical case, however, neither can be removed by a field redefinition. However, both functions are fully determined by our choice of solution with $\epsilon$, $s$, and $\tilde s$ constant. Differentiating Eq. (\ref{lagcuscs}) with respect to $X$ gives
\begin{equation}
{\cal L}_X = \frac{f\left(\phi\right)}{\sqrt{X}} + C = \frac{2 \epsilon}{s^2} \left(\frac{M_P}{\phi}\right)^2,
\end{equation}
where the right hand side is the solution (\ref{sqrtlagX}). Then
\begin{eqnarray}
f\left(\phi\right) &=& \sqrt{X} \left(\frac{2 \epsilon}{s^2}\right) \left(\frac{M_P}{\phi_0}\right)^2 \left[\left(\frac{\phi_0}{\phi}\right)^2 - 1\right]\cr
&=& \sqrt{X} \left(\frac{2 \epsilon}{s^2}\right) \left(\frac{M_P}{\phi_0}\right)^2 \left[c_S^2\left(\phi\right) - 1\right]
\end{eqnarray}
where for $2\tilde{s} = - s$, the expression (\ref{gamm}) for the
speed of sound becomes
\begin{equation}
\label{cscusc}
c_S\left(\phi\right)= \left(\frac{\phi_0}{\phi}\right).
\end{equation}
The Lagrangian (\ref{lagcuscs}) can then be written as
\begin{equation}
{\cal L} = X \left(\frac{2 \epsilon}{s^2}\right) \left(\frac{M_P}{\phi_0}\right)^2 \left[2 c_S^2\left(\phi\right) - 1\right] - V\left(\phi\right).
\end{equation}
The Hubble parameter (\ref{hubpar}) is given by
\begin{equation}
\label{eq:Hcusc}
H\left(\phi\right) = H_0 \left(\frac{\phi}{\phi_0}\right)^{\epsilon / s},
\end{equation}
and we can then express the kinetic term as a function of $\phi$ using Eq. (\ref{Xphi}):
\begin{equation}
X\left(\phi\right) = \frac{s^2}{2} H^2 \phi^2 = \frac{s^2}{2} \frac{ \phi_0^2 H^2\left(\phi\right)}{c_s^2\left(\phi\right)},
\end{equation}
The Lagrangian (\ref{lagcuscs}) can then be written entirely as a function of the field $\phi$,
\begin{equation}
{\cal L} = M_P^2 \epsilon H^2\left(\phi\right) \left[2 - \frac{1}{c_S^2\left(\phi\right)}\right] - V\left(\phi\right).
\end{equation}
The Hamilton-Jacobi Equation (\ref{hamjac1}) becomes:
\begin{eqnarray}
3 M_P^2 H^2 &=& 2 M_P^2 \epsilon H^2 - {\cal L}\cr
&=& V\left(\phi\right) + \frac{M_P^2 \epsilon H^2}{c_S^2},
\end{eqnarray}
and we have an expression for the potential $V\left(\phi\right)$,
\begin{equation}
V\left(\phi\right) = M_P^2 H^2\left(\phi\right) \left[3 - \frac{\epsilon}{c_S^2\left(\phi\right)}\right].
\end{equation}
The Hubble parameter $H\left(\phi\right)$ and the speed of sound $c_S\left(\phi\right)$ are given by Eqs. (\ref{eq:Hcusc}) and (\ref{cscusc}),
respectively. For $\phi / \phi_0 \ll 1$, the speed of sound is much greater than the speed of light, $c_S \gg 1$, and the potential is approximately
\begin{equation}
V\left(\phi\right) \simeq 3 M_P^2 H^2\left(\phi\right) = 3 M_P^2 H_0^2 \left(\frac{\phi}{\phi_0}\right)^{2 \epsilon / s},
\end{equation}
which can be recognized as a slow-roll-like solution dominated by
the potential $H^2 \simeq V^2 / {3 M_P^2}$. For $s < 0$, the
field is rolling away from the origin, and for $s < 1 - \epsilon$
the comoving acoustic horizon is shrinking and the solution is
tachyacoustic.

\subsection{{$\mathbf{n=3}$\bf{: The DBI model}}}
\label{subsec:dbi}

The case $n=3$, corresponds to $\tilde{s}=s$; then, from (\ref{defs}) and (\ref{defstil}), we find ${\cal{L}}_{X}=c_S^{-1}$. Eq. (\ref{difflagconst})
is then
\begin{equation}
c_S^2=\frac{1}{C {\cal{L}}_{X}^2},
\end{equation}
so that we can take $C = 1$ without loss of generality. Therefore, the Lagrangian assumes the well-known DBI form,
\begin{equation}
\label{lagDBIs}
{\cal{L}}\left(X,\phi\right)=-f^{-1}\left(\phi\right)\sqrt{1-f\left(\phi\right)X} + f^{-1}\left(\phi\right) - V\left(\phi\right).
\end{equation}
The DBI model with constant flow parameters is extensively discussed in Ref. \cite{Kinney:2007ag}, and the reader is referred this paper for further
details. For $\epsilon$ and $s$ constant, the functions $V$ and $f$ are fully determined and are given by
\begin{eqnarray}
\label{eq:potentials}
V\left(\phi\right) &=& 3 M_P^2 H^2\left(\phi\right) \left[1 - \left(\frac{2 \epsilon}{3}\right) \frac{1}{1 + c_S\left(\phi\right)}\right],\cr
f\left(\phi\right) &=& \left(\frac{1}{2 M_P^2 \epsilon}\right) \frac{1 - c_S^2\left(\phi\right)}{H^2\left(\phi\right) c_S\left(\phi\right)}.
\end{eqnarray}
The Hubble parameter and speed of sound are given by:
\begin{equation}
H\left(\phi\right) = H_0 \left(\frac{\phi}{\phi_0}\right)^{- 2 \epsilon / s},
\end{equation}
and
\begin{equation}
c_S\left(\phi\right) = \left(\frac{\phi}{\phi_0}\right)^2.
\end{equation}
DBI Lagrangians allow for either inflationary or tachyacoustic evolution \cite{Magueijo:2008sx}, depending on the values of $\epsilon$ and $s$. Note that for $c_S > 1$, the
function $f$ is negative, which has consequences for embedding such a model in string theory, which we discuss in Sec. \ref{sec:Conclusions}.

In this section, we have explicitly constructed Lagrangians,
including fully determined potentials, for which the flow
parameters are constant and the background evolution can be solved
exactly. For suitable choices of the flow parameters, the
evolution is tachyacoustic, {\it i.e.} with a growing comoving
Hubble horizon and a shrinking comoving acoustic horizon. In the
next section, we discuss the generation of curvature perturbations
at the acoustic horizon and show that such perturbations are
nearly scale-invariant, consistent with observation.

\section{Cosmological Perturbations for constant flow parameters}
\label{sec:Perturbations}

We can deal with cosmological perturbations in this general k-essence model with constant flow parameters in the same way as performed in
\cite{Kinney:2007ag}. Following the approach of Garriga and Mukhanov \cite{Garriga:1999vw} we start with the
perturbed Einstein equations,
\begin{eqnarray}
\label{eq:perteinst}
\frac{d}{dt}\left(\frac{\delta\phi}{\dot{\phi}}\right)&=&\Phi+\frac{2M_P^2c_S^2}{a^2(\rho+p)}\nabla^2\Phi\nonumber \\
\frac{d}{dt}\left(a\Phi\right)&=&\frac{a(\rho+p)}{2M_P^2}\left(\frac{\delta\phi}{\dot{\phi}}\right),
\end{eqnarray}
where $\Phi$ is the Bardeen potential and $\delta\phi$ is the
perturbation of the field $\phi$. Equations
(\ref{eq:perteinst}) can be cast into a more convenient form by
changing the perturbations $\Phi$ and $\delta\phi$ to the new
variables $\zeta$ and $\xi$ defined by
\begin{eqnarray}
\label{eq:newpert}
\xi&=&\frac{2M_P^2\Phi a}{H}\nonumber \\
\zeta&=&H\frac{\delta\phi}{\dot{\phi}}+\Phi,
\end{eqnarray}
so that the perturbed Einstein equations (\ref{eq:perteinst}) become
\begin{eqnarray}
\label{eq:newperteinst}
\dot\xi&=&\frac{a(\rho+p)}{H^2}\zeta,\nonumber \\
\dot\zeta&=&\frac{c_S^2H^2}{a^3(\rho+p)}\nabla^2\xi.
\end{eqnarray}
As usual, we introduce a new variable $z$ and the gauge-invariant Mukhanov potential $u$ as
\begin{equation}
\label{eq:defz}
z=\frac{a(\rho+p)^{1/2}}{c_SH},~~~~~ u=z\zeta;
\end{equation}
then, from (\ref{eq:newperteinst}) we derive the mode equation for $u(\tau)\propto u_k(\tau)\exp(i\mathbf{k}\cdot\mathbf{x})$, given by
\begin{equation}
\label{eq:modeeq}
u_k'' - \left[\left(c_S k\right)^2 + \frac{z''}{z}\right] u_k = 0,
\end{equation}
where a prime denotes a derivative with respect to conformal time,
$ds^2 = a^2\left(\tau\right) \left(d \tau^2 - d {\bf x}^2\right)$.
It is easy to show that the variable $z$, defined by
(\ref{eq:defz}) can be cast into the following form,
\begin{equation}
\label{newdefz}
z=-\frac{aM_P\sqrt{2\epsilon}}{c_S};
\end{equation}
then, using
\begin{equation}
\label{ddtau}
\frac{d}{d\tau}=-aH\frac{d}{dN},
\end{equation}
we can evaluate the ratio $z''/z$ in (\ref{eq:modeeq}) in terms of the flow parameters (\ref{defeps}-\ref{hflowpar}); the result is
\begin{equation}
\label{zF}
\frac{z''}{z} = a^2 H^2
\bar F\left(\epsilon,\tilde{\eta},s,\tilde{s},{}^2\lambda,{}^1\alpha,{}^1\beta\right),
\end{equation}
where
\begin{eqnarray}
\label{defF}\bar F &\equiv & 2+2\epsilon-3\tilde{\eta}-3s+\frac{3}{2}\tilde{s}+ 2\epsilon^2+\frac{5}{4}\tilde{s}^2-2s\tilde{s}\nonumber \\
&+&\tilde{\eta}^2+2\epsilon(\tilde{s}-s)+3\tilde{\eta} s-\frac{5}{2}\tilde{\eta}\tilde{s}
-4\tilde{\eta}\epsilon+ {}^2 \lambda\nonumber \\ &-&\frac{1}{2}\epsilon \left({}^1 \alpha\right)+\epsilon \left({}^1 \beta\right).
\end{eqnarray}
Next, it is convenient to change the conformal time, $\tau$, to the ratio of wavenumber to the sound horizon,
\begin{equation}
y \equiv \frac{c_Sk}{a H};
\end{equation}
then, conformal time derivatives switch to
\begin{equation}\label{dery1}
\frac{d}{d\tau} = - aH \left(1 - \epsilon -
s\right)y \frac{d}{dy},
\end{equation}
and
\begin{equation}\label{dery}
\frac{d^2}{d\tau^2} = a^2 H^2 \left[\left(1 - \epsilon -
s\right)^2 y^2 \frac{d^2}{dy^2} +
\bar G\left(\epsilon,\tilde{\eta},s,\tilde{s},{}^1 \alpha\right) y \frac{d}{d y}\right],
\end{equation}
where
\begin{eqnarray}
\label{defG}\bar G &\equiv& -s+\epsilon(2s+\tilde{s})+s(2s+\tilde{s})+2\epsilon^2-2\epsilon\tilde{\eta}-s\tilde{\eta}\nonumber \\ &-&\epsilon \left({}^1 \alpha\right).
\end{eqnarray}
It is important to stress that the functions $F$ and $G$ derived above hold in general; they reduce to the well known expressions in the DBI limit
\cite{Kinney:2007ag}, which, in this case, $s=\tilde{s}$ and ${}^1 \alpha={}^1 \beta=\rho$. Substituting (\ref{zF}) and (\ref{dery}) into the mode
equation (\ref{eq:modeeq}), we find
\begin{equation}
\label{exactmode} \left(1 - \epsilon -s\right)^2 y^2 \frac{d^2
u_k}{dy^2} + \bar G y \frac{d u_k}{d y} + \left[y^2 - \bar F\right] u_k = 0,
\end{equation}
which is an exact equation, without any assumption of slow-roll.

In the case where the flow parameters are constant, we can use the differential equations (\ref{flowequations}) to reduce the number of independent
parameters. We have
\begin{eqnarray}
\tilde{\eta} &=& \frac{1}{2}\left(2 \epsilon + \tilde{s}\right),\cr {}^2 \lambda
&=& \frac{1}{2}\left(2 \epsilon + \tilde{s}\right) \left(\epsilon +
\tilde{s}\right),\cr {}^1 \alpha &=&
\frac{s}{2 \epsilon}\left(2s +\tilde{s}\right),\cr
{}^1 \beta &=& \frac{3 {\tilde{s}}^2}{2 \epsilon};
\end{eqnarray}
then, substituting these values into expressions (\ref{defF}) and (\ref{defG}), we find, respectively,
\begin{eqnarray}
\label{Fflowc}\bar F &=& 2-\epsilon-3s+\frac{9}{4}\tilde{s}^2-\frac{3}{4}s\tilde{s}+\epsilon s-\frac{1}{2}s^2,
\end{eqnarray}
\begin{eqnarray}
\label{Gflowc}\bar G &=& s(-1+\epsilon+s).
\end{eqnarray}
It is important to notice that $\bar F$ is different from the
corresponding expression found in the DBI case
\cite{Kinney:2007ag}, since the gauge-dependent $\tilde{s}$ comes
into play. However, $\bar G$ is identical to its DBI analog, and
it is expected since basically it comes from the change of
variables $\tau\rightarrow y$, which depends solely on the
parameters $c_S$ and $H$, and not on ${\cal{L}}_{X}$. For constant
flow parameters we can solve equation (\ref{exactmode}) exactly,
and the solutions are given by
\begin{eqnarray}
\label{uksol}
u_{k}(y)&=&y^{\frac{1-\epsilon}{2(1-\epsilon-s)}}\left[c_{1}H_{\nu}^{(1)}\left(\frac{y}{1-\epsilon-s}\right)\right.\nonumber \\
&+&\left.c_{2}H_{\nu}^{(2)}\left(\frac{y}{1-\epsilon-s}\right)\right],
\end{eqnarray}
where $c_1$ and $c_2$ are constants, and $H_{\nu}^{(1)}$, $H_{\nu}^{(2)}$ are Hankel functions of first and second kind, respectively. The order $\nu$
of the Hankel function is given by
\begin{eqnarray}
\label{horder}
\nu^2=\frac{9-6\epsilon-12s+9\tilde{s}^2-3s\tilde{s}+4\epsilon s-2s^2+\epsilon^2}{4(1-\epsilon-s)^2};
\end{eqnarray}
next, using (\ref{solflowparN}), (\ref{ddtau})
and (\ref{dery1}) we find that
\begin{eqnarray}
\label{csy}
c_S\propto y^{s/(\epsilon+s-1)};
\end{eqnarray}
then, imposing the Bunch-Davies vacuum $c_2=0$, and normalizing the mode amplitudes by means of the canonical quantization condition
\begin{equation}
\label{Wronskian}
u_{k}^{*}\frac{du_{k}}{dy}-u_{k}\frac{du_{k}^{*}}{dy}=\frac{i}{c_{s}k(1-\epsilon-s)},
\end{equation}
we find
\begin{equation}
\label{uksolution}
u_{k}(y)=\frac{1}{2}\sqrt{\frac{\pi}{c_{s}k}}\sqrt{\frac{y}{1-\epsilon-s}}H_{\nu}\left(\frac{y}{1-\epsilon-s}\right),
\end{equation}
which differs from the DBI case only in the order of the Hankel function (\ref{horder}). In the small wavelength limit $y\rightarrow\infty$
the early-time behavior of $u_k$ will be identical to DBI \cite{Kinney:2007ag} for constant flow parameters
\begin{equation}
\label{uksmall}
u_{k}=\frac{1}{\sqrt{2c_{s}k}}e^{iy/(1-\epsilon-s)},
\end{equation}
whereas in the late-time behavior $y\rightarrow0$ the mode function behaves as
\begin{equation}
\label{uklong}
\left|u_{k}(y)\right| \rightarrow
2^{\nu-3/2}\frac{\Gamma(\nu)}{\Gamma(3/2)}(1-\epsilon-s)^{\nu-1/2}
\frac{y^{1/2-\nu}}{\sqrt{2 c_{s}k}}.
\end{equation}
From (\ref{uklong}) we can derive the expression for the scalar spectral index $n_s$. Using the definition of the power spectrum of curvature
perturbations
\begin{equation}
\label{curvspec1}
P_{\cal R}\left(k\right) =
\frac{k^3}{2 \pi^2} \left\vert\frac{u_k}{z}\right\vert^2,
\end{equation}
and substituting expressions (\ref{newdefz}) and (\ref{uklong}) into (\ref{curvspec1}), we find
\begin{equation}
\label{curvspec2}
P_{\mathcal{R}}=\frac{\left\vert f(\nu)\right\vert^2}{8\pi^2M_P^2}\frac{H^2}{c_S\epsilon}
\end{equation}
at horizon crossing, where $f(\nu)$ is a constant given by
\begin{equation}
\label{fnu}
f(\nu)=2^{\nu-3/2}\frac{\Gamma(\nu)}{\Gamma(3/2)}(1-\epsilon-s)^{\nu-1/2};
\end{equation}
then, from the definition of the scalar spectral index
\begin{equation}
\label{defns} n_{s}-1 \equiv
\frac{d\ln P_{\mathcal{R}}}{d\ln k},
\end{equation}
and using
\begin{equation}
\label{ddk} \frac{d}{d\ln
k}=-\left(\frac{1}{1-\epsilon-s}\right)\frac{d}{dN},
\end{equation}
we see that the spectral index $n_s$ assumes the form
\begin{equation}
\label{specindex} n_{s}=1-\frac{2\epsilon+s}{1-\epsilon-s},
\end{equation}
which does not depend on the gauge-dependent parameter $\tilde{s}$, and is identical to its DBI analog. This is expected since the power spectrum
evaluated at the horizon crossing, equation (\ref{curvspec2}), depends solely on $H$ and $c_S$, whose derivatives with respect to $N$ are related
to the gauge-invariant flow parameters $\epsilon$ and $s$. The scale-invariant limit is $s = - 2 \epsilon$.

\section{Conclusions}
\label{sec:Conclusions}

In this paper we have demonstrated that accelerated expansion or a
collapsing universe are not the only ways to dynamically generate
a scale-invariant spectrum of superhorizon curvature
perturbations. There is a third way: a superluminal acoustic cone
which is shrinking in comoving coordinates. Curvature
perturbations generated at the acoustic horizon are familiar from
inflationary scenarios based on non-canonical Lagrangians such as
$k$-inflation and DBI inflation. Such non-canonical Lagrangians
arise naturally in string theory.  However, in these scenarios,
{\it both} the Hubble horizon and the acoustic horizon are
shrinking in comoving units, and the acoustic horizon is typically
smaller than the Hubble horizon, {\it i.e.} $c_S < 1$. It is
natural to ask whether tachyacoustic models have a similar,
natural stringy embedding, especially since the DBI action
(\ref{eq:DBIlagrangian}) naturally admits tachyacoustic solutions.
Such an embedding is nontrivial, however, since the frequently
considered case of a 3+1 dimension d-brane evolving in a
higher-dimensional throat is ill-defined in the $c_S > 1$ limit.
To see this, consider the full ten-dimensional metric of throat
plus brane \cite{Klebanov:2000hb},
\begin{equation}
\label{eq:DBImetric}
ds^2_{10} = h^2\left(r\right) ds^2_4 + h^{-2}\left(r\right) \left(d r^2 + r^2 ds^2_{X_5}\right).
\end{equation}
The field $\phi$ is simply related to the coordinate in the throat
$r$ as $\phi = \sqrt{T_3} r$, where the brane tension  $T_3$
depends on the string scale $m_s$ and the string coupling $g_s$ as
\cite{Lidsey:2007gq}
\begin{equation}
T_3 = \frac{m_s^4}{\left(2 \pi\right)^3 g_s}.
\end{equation}
The Lagrangian for the field $\phi$ can be shown to be of the DBI form (\ref{eq:DBIlagrangian}),
where the inverse brane tension $f\left(\phi\right)$ is given in terms of the warp factor $h\left(\phi\right)$ by
\begin{equation}
\label{eq:warpfactor}
f\left(\phi\right) = \frac{1}{T_3 h^4\left(\phi\right)}.
\end{equation}
The problem is immediately evident: superluminal propagation $c_S
> 1$ requires $f < 0$, so that the factor $h^2\left(\phi\right)$
appearing in the metric (\ref{eq:DBImetric}) is imaginary, and the
metric is ill-defined. Therefore, although the DBI action itself
admits tachyacoustic solutions, this limit does not correspond to
a well-defined string solution. It is not clear whether or not
string manifolds exist which self-consistently admit solutions
with $c_S > 1$.

We calculate the scalar spectral index of perturbations for tachyacoustic solutions, and find
\begin{equation}
n_{s}=1-\frac{2\epsilon+s}{1-\epsilon-s}.
\end{equation}
Unlike inflationary models, radiation-dominated tachyacoustic
models do not require a period of explosive entropy production to
transition to a ``hot'' Big Bang cosmology. The early universe
must be scalar-field dominated, but the temperature of the
universe is not driven exponentially to zero, since the scalar has
a radiation equation of state at all times, and entropy density is
conserved (for any radiation component with density $\rho_\gamma$,
the ratio $\rho_\phi / \rho_{\gamma} = {\rm const}$). The scalar field $\phi$ must eventually decay to Standard Model degrees of freedom, but as long as this happens before primordial nucleosynthesis, the model will match observations. A slow or late decay of $\phi$ into other degrees of freedom would also suppress the production of unwanted relics such as monopoles or gravitinos. For
radiation-dominated tachyacoustic expansion with $\epsilon = 2$,
the spectral index is
\begin{equation}
n = 1 + \frac{4 + s}{1 + s},
\end{equation}
where we have $s < -3$ for a shrinking comoving acoustic cone. For
$s < -4$, the spectral index is blue, $n > 1$, which is ruled out
by observation. The WMAP $2 \sigma$ limit $n = 0.96 \pm 0.026$
\cite{Komatsu:2008hk} corresponds to $s = [-3.814,-3.959]$. Since
the Hubble horizon is growing in comoving units, no gravitational
wave modes are produced.

Tachyacoustic models are not a fully convincing alternative to
inflation, since they solve only the horizon problem and not the
flatness problem, and inflation solves both at once. However,
inflation has initial conditions problems of its own, in
particular the fact that the initial inflationary ``patch'' must
be larger than a horizon size for inflation to start
\cite{Vachaspati:1998dy}. Furthermore it has been shown that
inflationary spacetimes are in general geodesically
past-incomplete \cite{Borde:2001nh}. The initial conditions for
tachyacoustic cosmology are quite different than those for
inflation due to the presence of a true ``Big Bang'' singularity
at zero time. However, in this limit, the sound speed is {\it
infinite} and the tachyacoustic solution approaches an instanton.
To see this, examine the form of the DBI field Lagrangian
(\ref{eq:DBIlagrangian}) near the $\tau = 0$ boundary of a
tachyacoustic spacetime. From Eq. (\ref{eq:DBIsoundspeed}), the
$c_S \rightarrow \infty$ limit corresponds to $\phi \rightarrow
\infty$ and  $f\left(\phi\right) \dot\phi^2 \rightarrow -\infty$,
so that
\begin{equation}
{\mathcal L} \rightarrow \frac{\dot\phi}{\sqrt{\left\vert f\right\vert}} - V\left(\phi\right),
\end{equation}
where
\begin{equation}
\dot\phi \equiv \sqrt{g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi}.
\end{equation}
From Eq. (\ref{eq:potentials}), the asymptotic behavior of $V\left(\phi\right)$ and $f\left(\phi\right)$ are
\begin{eqnarray}
V\left(\phi\right) &\rightarrow& 3 M_P^2 H^2 \propto \phi^{-4 \epsilon / s},\cr
f\left(\phi\right) &\rightarrow& -\frac{1}{2 M_P^2 \epsilon} \frac{c_S}{H} \propto \phi^{2\left(1 + 2 \epsilon / s\right)}.
\end{eqnarray}
The scale-invariant limit $s = -2 \epsilon$ is especially interesting, since
\begin{equation}
\frac{1}{\sqrt{\left\vert f\right\vert}} \rightarrow \mu^2 = {\rm const.},
\end{equation}
and the Lagrangian takes the form
\begin{equation}
\label{eq:Lcuscuton}
{\mathcal L} \rightarrow \mu^2 \dot\phi - V\left(\phi\right),
\end{equation}
where $V\left(\phi\right) \propto \phi^2$. This can be identified
as exactly the ``cuscuton'' Lagrangian, suggested by Afshordi,
{\it et al.} as a candidate for Dark Energy
\cite{Mukhanov:2005bu,Afshordi:2006ad,Afshordi:2007yx}. Similarly, the $n = 0$
solution considered in Sec. \ref{sec:reconstruction} approaches a
cuscuton on the initial boundary surface.  The cuscuton is a
non-dynamical, instanton-like solution with infinite speed of
sound. Consider the action for the Lagrangian
(\ref{eq:Lcuscuton}),
\begin{eqnarray}
\label{eq:Scuscuton}
S_\phi &=& \int{d^4 x \sqrt{-g} \left[\mu^2 \dot\phi - V\left(\phi\right)\right]}\cr
&=& \mu^2 \int{dt \dot\phi} \int{d^3 x \sqrt{-g}} -  \int{d^4 x \sqrt{-g} V\left(\phi\right)}\cr
&=&  \mu^2 \int{d\phi \Sigma\left(\phi\right)} - \int{d^4 x \sqrt{-g} V\left(\phi\right)},
\end{eqnarray}
where $\Sigma\left(\phi\right)$ is the volume of a constant-$\phi$
hypersurface in the spacetime. The classical solutions to the
cuscuton action are constant mean curvature hypersurfaces,
analogous to soap bubbles \cite{Afshordi:2006ad}. It is
interesting to speculate that this property of the cuscuton action
may provide a self-consistent cosmological boundary condition, or
(even more speculatively) be useful as a solution to the
cosmological flatness problem. A full analysis, however, would
require inclusion of the gravitational action and solution in a
Wheeler-De Witt framework, or perhaps an embedding of the model
in string theory or an alternate gravity theory such as Horava-Lifshitz \cite{Piao:2009ax,Afshordi:2009tt}.
This is the subject of future work.

redpajama_set_name: RedPajamaArXiv


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Offering and securing a variety of sponsorship opportunities for a fundraising event can be one of the most important measurements of success. Yet, soliciting and securing sponsorships is one of the most intimidating parts of the job.
We often feel nervous making the ask—coming right out and putting a dollar value on our request for support.
Before you go into your next sponsorship meeting, consider the additional value to offer potential sponsors with online and customized sponsorship opportunities.
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Remember, your nonprofit has a huge amount of value to offer in exchange for sponsorship dollars. You're not seeking charity; you're seeking a mutually beneficial partnership. You're providing marketing, exposure to your audience, and the opportunity to build trust within the community.
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text:
Home / Indies / WSUWrestling.com News Update For 12/19
WSUWrestling.com News Update For 12/19
By Brad Dykens on December 20, 2009
Make sure to check out WSUWrestling.com & DOIVideo.com later this week for three brand new DVD Releases!!!
Women Superstars Uncensored (WSU) will return with another full fledged double DVD taping on March 6th, for our Third Anniversary Show at the ACE Arena (725 Sip Street, Union City, NJ). WSU, the innovators of womens wrestling, will be doing something new for this set of tapings, as we will have a mini-convention, a meet & greet, the WSU Class of 2010 Hall of Fame Ceremony & the Third Anniversary Show. This will be a great celebration for fans of WSU and womens wrestling alike.
We are still working on complete details and getting everything organized. We hope to have everything 100% announced and confirmed by January 1st. We are looking to run a mini-convention and a meet/greet with the WSU roster & several big names from womens wrestling. Many of our fans have asked for more time to meet the girls of WSU during intermission, so we are hoping to give fans good ample time to converse with their favorite WSU wrestlers with this meet/greet & convention. This will take place in the afternoon. After a break, we will open up with the WSU Hall of Fame Ceremony, where we will be honoring several names in womens wrestling. We will then take another break, and then open with our first of two DVD Tapings. Again, we are still getting everything organized, and we hope to have everything set, talent announcements and price/tickets announced shortly.
To make it clear, we will be taping 3 different DVDs to celebrate our Three Year Anniversary. We have many top matches signed already and many feuds will take center stage at the Anniversary show. The first DVD will cover the first half of the show, with the main event of Jazz vs Amber O'Neal. The second DVD will cover the second half of the show, with the main event of Mercedes Martinez vs Rain. The third DVD will be footage of t he convention, the complete Hall of Fame ceremony and WSU title matches that take place from January 2010-March 2010. (We hope to have a full announcement on upcoming matches shortly.)
So far, here are the matches & talent announced thus far for 3/6 at the ACE Arena in Union City, NJ:
Matches set for WSU's Third Anniversary show thus far are:
WSU World Title Match
Uncensored Rules
(c) Mercedes Martinez vs Rain
WSU Spirit Championship Match
(c) Alicia vs Brittney Savage w/Rick Cataldo
*As part of the re-match stipulation, if Alicia wins, she gets 5 minutes alone with Rick Cataldo.
Someone's Undefeated Streak Will End
Jazz vs Amber O'Neal
Almost Two Years in the Making
Battle of Two Former World Champions
Angel Orsini vs Nikki Roxx
Submission Match
The Match To Finally End The Feud
Jana vs Latasha
Also appearing are Jessicka Havok, Angelina Love, Jennifer Cruz, Destiny, Cindy Rogers, The Return of Alere Little Feather & many more to be announced in upcoming weeks.
Just Added: MOLLY HOLLY returns to WSU
WSU will also return with our big brother promotion NWS on 4/3 and 4/4 for the 2010 Women's J-Cup & King/Queen Tournaments. More info on those shows to be announced at a later tonight.
WSU is taking January/February off from full length events (But we will be having title matches/On The Road matches at other events) in order to prepare for what will be the biggest womens wrestling show of 2010. As we write this news release, the northeast is flooded with snow, and we can not take a chance of being hit with a snowstorm for this event. In previous years, we have had to run shows in bad weather in the winter time and we don't feel it is fair to our fans or the wrestlers to put anyone in hazardous traveling conditions, which is why we are taking Jan/Feb off. We are hoping that the weather gods will cooperate with us in March and we hope to see you all there for what will be one of the most stacked & packed womens wrestling show of the year.
We will have an updated and more exact press release after the new year. Happy Holidays.
WSUWrestling.com
← Previous Story IWC A Call to Arms – A Weekend of Change
Next Story → WSU releases three brand new DVD's

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: filter data with comma with where clause firestore firebase angular i have field in firestore in which data is stored like this
asdasd, chocolates

i want to filter data with these tags for example if i send paramter to link like this http://localhost:4200/blogs/tagBlogList/chocolates
i need to get all data who have 'chocolates' tags
here is my code that i am trying to get data
this.featureBlogCollection = this.afs.collection('posts', ref => ref.where("category","array-contains-any",[category]))

    //this.blogDoc = this.afs.doc<Blog>(`posts/${category}`)

    return this.featureBlogCollection.valueChanges()

but i am getting empty data with this code

A: You need to store your tags as an array, for your code to work correctly :
Instead of saving:
asdasd, chocolates

You need to save it as
['asdasd', 'chocolates']

Your code currently is search for the value "chocolates", but it's only finding "asdasd, chocolates" which isn't a match.
You can store as an array by using plain javascript:
// Code just before saving your document
let categoriesArray = categoriesString.split(',');
// Optional: Trim whitespace
categoriesArray = categoriesArray.map((item) => item.trim());
// Attach categories to your entry and save it.

For stronger searching you'll have to wire up a 3rd party full text search service of your choice, otherwise, this should work.

redpajama_set_name: RedPajamaStackExchange


What's it like being part of a development team like the DeltaWing?
How do you adjust from an open-wheel car to the DeltaWing?
What is your preference for a track?
What are some of your hobbies when you're not in a race car?


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The Adventure Club contributes to the educational mission of the university by offering learning opportunities to the students who pursue leadership and try to make the impossible possible and remain indomitable in impediments.
Our Adventures club's chief purpose is to engage students in adventure experiences for escape, education and enjoyment. We're here to enhance the student experience. Using adventure in its many forms, we aim to expand and diversify students' experiences, and provide experimental education that accentuates lessons learned in the future.
Our activities include social outings like annual tours, charity events, concerts and free medical camps in remote areas. The Club also organizes annual picnic for the interest of students and members of the faculty and management from various walks of life. It helps the students to visit the historical places to highlight cultural profiles of our country. These adventurous activities will develop our student's ability at an individual and personal level, and allow them to take on greater challenges socially and academically.


text:
Thief who robbed Manchester Arena bombing victims being hunted by police after prison release
Chris Parker was first hailed a hero in the aftermath of the Manchester Arena atrocity after he claimed he ran inside the venue to help immediately after the blast
Sam Yarwood
Brett Gibbons
A homeless man jailed for stealing from victims of an terror attack is being hunted by police breaching the conditions of his release from prison.
Chris Parker was hailed a hero in the aftermath of the Manchester Arena atrocity, with strangers donating more than £50,000 to get him off the streets after he claimed he ran inside the venue to help immediately after the blast.
But in reality he was a heartless thief who stole from the stricken and desperate, the Manchester Evening News reports.
CCTV recovered by police showed Parker wandering around the foyer area of the Arena amid the dying and injured before looking through bags.
(Image: PA)
He was arrested, and in January 2018 admitted to stealing a purse belonging to Pauline Healey, whose 14-year-old granddaughter Sorrell Leczkowski was killed in the atrocity.
The then 33-year-old also stole an iPhone from a teenager who cannot be named for legal reasons and took photos of the wounded.
Prosecutors said that Parker did provide 'some limited assistance' to the injured, but Manchester Crown Court heard he 'equally' took the opportunity to steal from people.
As well as admitting two counts of theft, he also pleaded guilty to one count of fraud, after using Mrs Healey's stolen bank card in a McDonald's in Manchester between May 23 and May 26 - days after the attack that claimed the lives of 22 people.
Travelodge reveals bizarre items left behind in Birmingham - including Jaguar car and Labradoodle
Favourite holiday spots among world's most dangerous places - is your destination on the list?
Following his appearance in court, the GoFundMe page was shut down and the thousands of people who donated were able to request a refund.
Parker was sentenced later that month to four years and three months behind bars.
It is not clear exactly when he was released from prison, however Greater Manchester Police has confirmed the now 35-year-old is wanted on recall to jail.
A spokesman for the force said: "We can confirm Parker breached his license conditions and is wanted on recall to prison."
Anyone who sees Parker or knows of his whereabouts is asked to contact police on 101.
CourtsHarvey Weinstein branded 'rapist' as he faces jury in landmark moment for MeToo movement"He was not just a titan in Hollywood. He was a rapist," prosecutor Meghan Hast told seven male and five female jurors
RugeleyPublic services to take hit as Amazon wins £3.2m tax wrangle over giant Rugeley depotThe tech giant says it has invested more than £9billion in the UK since 2010 and created 2,500 jobs in 2018 alone
Aston Villa FCJack Grealish says what every Aston Villa is thinking after that Watford victoryAston Villa 2 - 1 Watford | Here's how Grealish, Mings, Smith and Konsa reacted to last night's events


text:
Bellevue High School
'Into the Woods' follows journey of self-discovery
Cary Ashby • Apr 4, 2019 at 12:00 PM
BELLEVUE — Bellevue High School students Joshua Higgins and Katie Jo Porter are playing a married couple for the second time in row.
Higgins, a sophomore, plays the baker in "Into the Woods" while Porter, a senior, plays his wife. The 25-member cast will perform the musical at 7 p.m. Friday and Saturday and again at 3 p.m. Sunday at Bellevue Elementary School.
The baker, while on a journey of self-discovery, goes from being self-centered and "full of himself" to discovering important keys to a marriage, Higgins said.
"He starts at this low point," added the son of Dan and Liesl. "Life is now just routine, kinda disappointing day after disappointing day. … At the end of the show — I don't want to give too much way — he learns kinda how to become a strong individual."
Higgins has performed in the last two fall musicals at Bellevue. He was in "All Shook Up" most recently and as a freshman, he performed in "The Hunchback of Notre Dame."
"Art isn't the best of the best; art is what you choose to make of what you have and give that out to the world. So I think we're doing a really nice job here at Bellevue of making art," the sophomore said.
The Bellevue cast recently had been rehearsing two shows since the students were preparing "All Shook Up" for the Ohio Thespian State Conference.
"The balance has gone really, really well. Our casts in both shows have been amazing. Our director (Angie Bickhart-Sommers) is just amazing, casting people (who) fit well in our roles," Higgins said. "That level of togetherness and community we have as a cast is amazing."
For "All Shook Up," he and Porter played a couple who fell in love. Higgins said he and his fellow cast member went from portraying a relationship with "heightened emotion and a little bit awkward passion" in the fall show to the stagnant marriage of "Into the Woods."
"As we go into the woods, the baker realizes he needs his wife there," he added.
Freshman Amber Bowering plays the witch, who visits the couple at their home early in the show.
Bowering — with a big smile — said she has to channel "my inner high-schooler," a student who is "mean, really mean." The teenager said she hunches over while in character, throws in some sarcasm and acts like a "know-it-all."
"Then I kinda put everyone in their place and tell them, 'I'm better than you,'" added the daughter of Angela and Jason. "A lot of attitude."
Embodying the witch has been a challenge.
"Because at some points in time, I want to come back to who I really am; I'm very bubbly and I like to dance around," Bowering said. "I have a lot of energy as a person, so I just kinda like, go on stage and have as much fun as I can. … You've just gotta put the energy into the songs."
Tackling Stephen Sondheim's music has been a challenge.
"He doesn't write music for middle-schoolers; this is just really, really challenging stuff," Higgins said.
The baker eventually meets Jack (junior Julia Ochenduski) from "Jack and the Beanstalk."
"His family is very poor and he has to sell his best friend, Milky White. His mom is making him sell the cow," said Ochenduski, whose character faces a lot of pressure.
"He doesn't want to do it, so he's very frustrated this whole time. He's like, 'I don't want to give up my cow.' He ends up running into the baker and making a really bad decision of selling his cow for some beans."
Ochenduski mostly was comfortable singing Jack's songs because she is a tenor, the higher range for a male.
"So a lot of it fit pretty perfectly in my vocal range," she said.
"All of the music numbers blend everything together. We start with three separate stories of 'Cinderella,' 'Jack and the Beanstalk' and the baker and his wife and they kinda transition into they're all going into the woods for a different (purpose)," the teenager added. "It all just kinda amazingly squashes itself; I'm not sure how (Sondheim) did it."
When Jack appears later in the musical, "he actually causes quite a bit of trouble that kinda moves the story plot along," Ochenduski said. "Into the Woods" then follows "Jack and the Beanstalk," with the giants pursuing Jack.
"They come down and wreak all sorts of havoc because he stole all of their stuff," said Ochenduski, the daughter of John and Nicole.
Before this production, Ochenduski saw the film version of "Into the Woods." Later, she bought a copy of yet another version, from Broadway, when Family Video closed in Bellevue. She said the Broadway rendition was funnier and not "as dark and serious" as the 2014 movie.
"Ours is very funny — a lot of comedy, a lot of craziness," she added.
Bowering, who has been doing theater for seven years, including performances at the Bellevue Society for the Arts, provided several reasons to see the Bellevue production of "Into the Woods."
"Why would someone want to see this musical? Because not only do they get to see their friends, but they get to see them perform and you get to see people have … a really good time embodying someone that they don't always get to play — someone totally, completely opposite from who they really are."


Episode 30, "Assault on South Headquarters"
Synopsis: At a refugee camp, Yoki spots Scar. Scar and Ishbalans are allowed to stay there with the other people. Yoki runs off to tell the military, hoping to gain his position back. Back in Dublith, Ed and Al chase the mysterious boy, but are approached by a strange lizard-man. Lieutenant Colonel Frank Archer and Armstrong arrive in Dublith, looking to find Izumi. After chasing off the lizard-man, the brothers find the boy on a rooftop. Edward figures that if his arm was brought back from the Gate, that Al's body could be, too. Armstrong captures the boy for Archer, who reveals the homunculi's identifying mark, the tatoo of the ouroborus. Juliet Douglas recieves word from Yoki about Scar. The lizard-man calls Greed, who has Tucker and Kimbley with him. Greed dispatches Kimbley to retrieve the boy. Izumi angrily chides the Elrics, but collapses and is taken to the doctor. The doctor reveals that Izumi is missing several internal organs, making the brothers realize she committed the same taboo they did. Izumi escapes with her husband and walks into the Southern Headquarters, using alchemy to make her way to the mysterious boy. Sig reveals to Winry that Izumi sacrificed her organs in a bid to bring their son back to life. Ed and Al confront Izumi inside, but then Kimbley and the lizard-man arrive, as well as the Fuhrer (Envy in disguise). In the refugee camp, Scar investigates the home of an Ishbalan outcast, and finds the same tattoo he has on his arm on the wall.
Yikes. There is a lot happening in this episode. A whole lot. It hardly lets up.
We're formally introduced to the black hole of charisma that is Lieutenant Colonel Frank Archer, the douchebaggier equivalent of Roy Mustang, if he wasn't an alchemist. Instead, Archer has a pet Armstrong for a while. He's really not a remarkable character. The only interesting thing about Archer at this point is that Armstrong is working under him, and becomes an obstacle to the Elric brothers' goals.
When on the roof, Edward figures that since the child has his arm and leg, it might be possible to bring Al's full body back out of the Gate. And we got this without two hours of chatchatchatchatchat about nutrients and memories.
One of my favorite Greed scenes is in this episode, where he's just sort of hanging out with some women and his gang at the Devil's Nest. A hive of scum and villainy, that place. It's scenes like this that made me really like Greed. His jaunty smugness and off-beat attitude really makes him a colorful character. Although, it's a bit weird that he actually keeps that bag around, especially considering what we later learn about it.
My favorite part of this episode is Izumi's assault on the Southern Headquarters (title drop!). She just walks right in and completely dominates the whole place, showing what a true badass she is. Although a lot of it is that they're completely caught off guard, not expecting some woman to just waltz in and grab the boy. And the scene manages to tug on the heart strings, too, since Sig tells Winry about Izumi's child over the footage of Izumi retrieving the child. My only question is, where exactly is Southern HQ? It must be pretty close to Dublith for them to get there so fast.
Things really get busy at the end of the episode with Izumi being confronted by Ed and Al, Armstrong and Archer, Kimbley and Bido, and a Bradley-disguised Envy. But the real conclusion of the episode has Scar stumbling on the design of his tattoo in the shack of an Ishbalan outcast. This episode is packed, but it doesn't feel too frantic, and is one of my favorite in the series.


In a remote mountain village far away in Northern Japan, Ashitaka, the last young warrior of the dying Emishi clan, is forced to kill a monster to protect his village. Too late, he discovers that the boar-like creature he killed is a protector-god of the forest. The god had been driven insane by a small iron pellet, found deep inside of its body. Although the villagers immediately lay the body to rest, they cannot escape the dying creature's words. The day of the human is coming to a close.
In killing the demon boar, Ashitaka has brought upon himself a curse, its sign a twisted scar on his right forearm that is slowly spreading. The village elders can do nothing for him, and the only chance Ashitaka has means he has to be banished from his village, never to set foot there again.
Ashitaka journeys to the land of the iron forging Tatara clan where he hopes to uncover the mystery of the curse before it takes his young life. Along the way, he becomes involved in the bitter fight between two warring factions of humans and a race of forest-gods trapped in the middle.
During one battle, Ashitaka sees San, the Princess Mononoke, a young woman raised by wolves, who is prepared to die to defeat the humans. She and her wolves will stop at nothing to kill the Tatara village leader, the regal Lady Eboshi. Against his will, Ashitaka is drawn into the struggle between man and nature, trying to persuade the two sides to find another way, one that will halt the endless circle of killing. He seeks a middle ground where forest creatures and humans can live in the world in peace together.
But man's greed truly knows no bounds.
Princess Mononoke is perhaps one of Hayao Miyazaki's strongest pieces of work. An epic tale, Princess Mononoke details the ongoing struggle between nature and mankind. The central character Ashitaka serves to try and show us that perhaps there is a way in which nature and humans can co-exist without destroying each other. Moral lesson aside, the storytelling in Princess Mononoke is nothing short of superb. It is a simple story, but crafted so elegantly and potently, that one cannot help but receive the message put forward by Miyazaki.
The animation is lush and vibrant, with the characters and locations all colourfully rendered. The movements are very fluid, and the attention to detail is astonishing. The designs of the forest spirits are very well conceived, something we have come to expect from Miyazaki.
The soundtrack is filled with hauntingly beautiful classical music scores, which adds beautifully to the unfolding story. The voices are spot on, and the sound-effects for the various creatures is nothing short of superb.
Seeing Princess Mononoke's success in Japan, the second movie to ever break $150 million at the Japanese box office (the other being 'Titanic'), Miramax Films decided to release a dubbed version to Western audiences. The film was very well received and paved the way for such future cinema releases of Japanese animated feature films.
Overall, Princess Mononoke stands out as my personal favourite piece of anime of all time. I cannot find any fault with it at all. The story, art and soundtrack are sublime. This is anime at its zenith. If you have not yet seen Princess Mononoke, then I suggest you make a plan to see it as soon as possible.


text:
package cz.zcu.kiv.eeg.KIMBridge.repository.google;

import com.google.api.client.http.HttpResponse;
import com.google.api.services.drive.model.File;
import cz.zcu.kiv.eeg.KIMBridge.connectors.google.DriveConnector;
import cz.zcu.kiv.eeg.KIMBridge.logging.ILogger;
import cz.zcu.kiv.eeg.KIMBridge.repository.IBinaryDocument;

import java.io.IOException;
import java.io.InputStream;

/**
 * Document stored in Google Drive. Does not support document updates.
 *
 * Document data is downloaded each time the {@code getData()} method is invoked.
 * @author Jan Smitka <jan@smitka.org>
 */
public class DriveDocument implements IBinaryDocument {
	private DriveConnector drive;

	private File file;

	private ILogger logger;

	/**
	 * Creates the document from Google Drive File metadata.
	 * @param connector Google Drive connector.
	 * @param fileMetadata File metadata.
	 */
	public DriveDocument(DriveConnector connector, File fileMetadata) {
		drive = connector;
		file = fileMetadata;
	}

	/**
	 * Sets the logger.
	 * @param logger Logger.
	 */
	public void setLogger(ILogger logger) {
		this.logger = logger;
	}

	/**
	 * Gets the random dummy document ID.
	 * @return See {@code http://xkcd.com/221/}.
	 */
	@Override
	public long getId() {
		return 4;
	}

	/**
	 * Is this a new document?
	 * @return Always {@code true}.
	 */
	@Override
	public boolean isNew() {
		return true;
	}

	/**
	 * Gets the document title.
	 * @return Original filename.
	 */
	@Override
	public String getTitle() {
		return file.getOriginalFilename();
	}

	/**
	 * Gets the document URL.
	 * @return URL in Google Drive.
	 */
	@Override
	public String getUrl() {
		return file.getAlternateLink();
	}

	/**
	 * Gets the document extension.
	 * @return Document file extension.
	 */
	@Override
	public String getExtension() {
		return file.getFileExtension();
	}

	/**
	 * Downloads the document data.
	 * @return Raw file data.
	 * @throws IOException when the data cannot be downloaded.
	 */
	@Override
	public byte[] getData() throws IOException {
		logger.logMessage("Downloading %s", file.getOriginalFilename());
		HttpResponse response = drive.sendDownloadFileRequest(file);
		int length = response.getHeaders().getContentLength().intValue();
		int offset = 0;
		byte[] data = new byte[length];
		InputStream stream = response.getContent();
		while (offset < length) {
			offset += stream.read(data, offset, length - offset);
		}
		stream.close();
		response.disconnect();
		logger.logMessage("%d bytes downloaded", length);
		return data;
	}
}

redpajama_set_name: RedPajamaGithub


text:
Probate Terms Defined
Accountings -- Reports that contain the annual and final fiscal reports showing receipts and disbursements with date and purpose.
Application -- a written request to the probate court for an order or statement of informal probate or appointment.
Beneficiary -- as it relates to trust beneficiaries, includes a person who has any present or future interest, vested or contingent, and also includes the owner of an interest by assignment or other transfer and, as it relates to a charitable trust, includes any person entitled to enforce the trust.
Child -- any individual entitled to take as a child by intestate succession from the parent whose relationship is involved and excludes any person who is only a stepchild, a foster child, a grandchild, or any more remote descendant.
Claims -- liabilities of the decedent or protected person whether arising in contract, in tort, or otherwise, and liabilities of the estate which arise at or after the death of the decedent or after the appointment of a conservator, including funeral expenses and expenses of administration. The term does not include estate or inheritance taxes, or demands or disputes regarding title of a decedent or protected person to specific assets alleged to be included in the estate.
Court -- the court or branch having jurisdiction in matters.
Conservator -- a person who is appointed by a court to manage the estate of a protected person.
Devise -- when used as a noun, means a testamentary disposition of real or personal property, including both devise and bequest as formerly used, and when used as a verb, means to dispose of real or personal property by will.
Devisee -- means any person designated in a will to receive a devise. In the case of a devise to an existing trust or trustee, or to a trustee on trust described by will, the trust or trustee is the devisee and the beneficiaries are not devisees.
Disability -- means cause for a protective order as described by § 62-5-401 which states Upon petition and after notice and hearing in accordance with the provisions of this part, the court may appoint a conservator or make other protective order for cause as follows:
(1) Appointment of a conservator or other protective order may be made in relation to the estate and affairs of a minor if the court determines that a minor owns money or property that requires management or protection which cannot otherwise be provided, has or may have business affairs which may be jeopardized or prevented by his minority, or that funds are needed for his support and education and that protection is necessary or desirable to obtain or provide funds,
(2) Appointment of a conservator or other protective order may be made in relation to the estate and affairs of a person if the court determines that (i) the person is unable to manage his property and affairs effectively for reasons such as mental illness, mental deficiency, physical illness or disability, advanced age, chronic use of drugs, chronic intoxication, confinement, detention by a foreign power, or disappearance; and (ii) the person has property which will be wasted or dissipated unless proper management is provided, or that funds are needed for the support, care, and welfare of the person or those entitled to be supported by him and that protection is necessary or desirable to obtain or provide funds.
Distributee -- means any person who has received property of a decedent from his personal representative other than as creditor or purchaser. A testamentary trustee is a distributee only to the extent of distributed assets or increment thereto remaining in his hands. A beneficiary of a testamentary trust to whom the trustee has distributed property received from a personal representative is a distributee of the personal representative. For purposed of this provision, "testamentary trustee" includes a trustee to whom assets are transferred by will, to the extent of the devised assets.
Estate -- includes the property of the decedent, trust, or other person whose affairs are subject to this Code as originally constituted and as it exists from time to time during administration
Exempt Property -- means that property of a decedent's estate which is described in § 62-2-401 which states The surviving spouse of a decedent who was domiciled in this State is entitled from the estate to a value not exceeding twenty-five thousand dollars in excess of any security interests therein in household furniture, automobiles, furnishings, appliances, and personal effects. If there is no surviving spouse, minor or dependent children of the decedent are entitled jointly to the same value. If encumbered chattels are selected and if the value in excess of security interests, plus that of other exempt property, is less than twenty-five thousand dollars, or if there is not twenty- five thousand dollars worth of exempt property in the estate, the spouse or children are entitled to other assets of the estate, if any, to the extent necessary to make up the twenty- five thousand dollar value. Rights to exempt property and assets needed to make up a deficiency of exempt property have priority over all claims against the estate except claims described in Section 62-3-805(a)(1). These rights are in addition to any right of homestead and personal property exemption otherwise granted by law but are chargeable against and not in addition to any benefit or share passing to the surviving spouse or children by the will of the decedent unless otherwise provided, by intestate succession, or by the elective share. Any surviving spouse or minor or dependent children of the decedent who fails to survive the decedent by one hundred twenty hours is deemed to have predeceased the decedent for purposes of this section.
Expense of Administration -- includes commissions of personal representatives, fees and disbursements of attorneys, fees of appraisers, and such other expenses that are reasonably incurred in the administration of the estate.
Fair market value -- is the price that property would sell for on the open market that would be agreed on between a willing buyer and a willing seller, with neither being required to act, and both having a reasonable knowledge of the relevant facts.
Fiduciary -- includes personal representative, guardian, conservator, and trustee.
Foreign Personal Representative -- means a personal representative of another jurisdiction.
Formal proceedings -- means actions commenced by the filing of a summons and petition with the probate court and service of the summons and petition upon the interested persons. Formal proceedings are governed by and subject to the rules of civil procedure adopted for circuit courts and other rules of procedure in this title.
Guardian -- means a person appointed by the court as guardian, but excludes one who is a guardian ad litem.
Guardianship proceeding -- is as defined in § 62-5-101 (5) is a formal proceeding under the provisions of Part 3 of Article 5 (Section 62-5-301, et seq.) to determine if a person is an incapacitated person, or to appoint a guardian for an incapacitated person.
General power of appointment -- means any power that would cause income to be taxed to the fiduciary in his individual capacity under Section 678 of the Internal Revenue Code and any power that would be a general power of appointment, in whole or in part, under Section 2041(a)(2) or 2514(c) of the Internal Revenue Code.
Heirs -- means those persons, including the surviving spouse, who are entitled under the statute of intestate succession to the property of a decedent.
Incapacitated person -- is as defined in § 62-5-101 (1) which states an "incapacitated person" means any person who is impaired by reason of mental illness, mental deficiency, physical illness or disability, advanced age, chronic use of drugs, chronic intoxication, or other cause (except minority) to the extent that he lacks sufficient understanding or capacity to make or communicate responsible decisions concerning his person or property.
Informal proceedings -- means those commenced by application and conducted without notice to interested persons by the court for probate of a will or appointment of a personal representative. Informal proceedings are not governed by or subject to the rules of civil procedure adopted for the circuit court.
Interested person -- includes heirs, devisees, children, spouses, creditors, beneficiaries, and any others having a property right in or claim against a trust estate or the estate of a decedent, ward, or protected person which may be affected by the proceeding. It also includes persons having priority for appointment as personal representative and other fiduciaries representing interested persons. The meaning as it relates to particular persons may vary from time to time and must be determined according to the particular purposes of, and matter involved in, any proceeding.
Issue of a person -- means all his lineal descendants whether natural or adoptive of all generations, with the relationship of parent and child at each generation being determined by the definitions of child and parent contained in this Code.
Lease -- includes an oil, gas, or other mineral lease.
Letters -- includes letters testamentary, letters of guardianship, letters of administration, and letters of conservatorship
Minor -- means a person who is under eighteen years of age, excluding a person under the age of eighteen who is married or emancipated as decreed by the family court
Mortgage -- means any conveyance, agreement, or arrangement in which real property is used as security.
Nonresident decedent -- means a decedent who was domiciled in another jurisdiction at the time of his death.
Organization -- includes a corporation, government or governmental subdivision or agency, business trust, estate, trust, partnership or association, two or more persons having a joint or common interest, or any other legal entity.
Parent -- includes any person entitled to take, or who would be entitled to take if the child died without a will, as a parent under this Code by intestate succession from the child whose relationship is in question and excludes any person who is only a stepparent, foster parent, or grandparent.
Person -- means an individual, corporation, business trust, estate, trust, partnership, limited liability company, association, joint venture, government or governmental subdivision, agency, or instrumentality, public corporation, or any other legal or commercial entity. .
Personal representative -- includes executor, administrator, successor personal representative, special administrator, and persons who perform substantially the same function under the law governing their status. "General personal representative" excludes special administrator.
Petition -- means a complaint as defined in the rules of civil procedure adopted for the circuit court. A petition requires a summons and is governed by and subject to the rules of civil procedure adopted for the circuit court and other rules of procedure in this title.
Probate estate -- means the decedent's property passing under the decedent's will plus the decedent's property passing by intestacy.
Proceeding -- includes action at law and suit in equity.
Property -- includes both real and personal property or any interest therein and means anything that may be the subject of ownership.
Protected person -- is as defined in § 62-5-101 (3) which states a "protected person" is a minor or incapacitated person for whom a conservator has been appointed or other protective order has been made.
Protective proceeding -- is as defined in § 62-5-101 (2) which states a "protective proceeding" is a proceeding under the provisions of Section 62-5-401 to determine if a person is an incapacitated person, or to secure the administration of the estates of incapacitated persons or minors.
SCACR -- means the South Carolina Appellate Court Rules.
Security -- includes any note, stock, treasury stock, bond, debenture, evidence of indebtedness, certificate of interest, or participation in an oil, gas, or mining title or lease or in payments out of production under such a title or lease, collateral trust certificate, transferable share, voting trust certificate or, in general, any interest or instrument commonly known as a security or any certificate of interest or participation, any temporary or interim certificate, receipt or certificate of deposit for, or any warrant or right to subscribe to or purchase, any of the foregoing.
Security interest -- means any conveyance, agreement, or arrangement in which personal property is used as security.
Settlement -- in reference to a decedent's estate includes the full process of administration, distribution, and closing.
Special administrator -- means a personal representative as described by 62-3-614 through 62-3-618.
State -- means a state of the United States, the District of Columbia, Puerto Rico, the United States Virgin Islands, a federally recognized Indian tribe, or any territory or insular possession subject to the jurisdiction of the United States.
Successor personal representative -- means a personal representative, other than a special administrator, who is appointed to succeed a previously appointed personal representative.
Successors -- means those persons, other than creditors, who are entitled to property of a decedent under his will or this Code.
Testacy -- proceeding means a formal proceeding to establish a will or determine intestacy.
Testator -- A person who has made a legally valid will before death.
Trust -- includes any express trust, private or charitable, with additions thereto, wherever and however created. It also includes a trust created or determined by judgment or decree under which the trust is to be administered in the manner of an express trust. "Trust" excludes other constructive trusts, and it excludes resulting trusts, conservatorships, personal representative, trust accounts as defined in Article 6 (Sections 62-6-101 et seq.), custodial arrangements pursuant to the South Carolina Uniform Gifts to Minors Act, Section 20-7-140 et seq., of the 1976 Code, business trusts providing for certificates to be issued to beneficiaries, common trust funds, voting trust, security arrangements, liquidation trusts, and trusts for the primary purpose of paying debts, dividends, interest, salaries, wages, profits, pensions, or employee benefits of any kind, and any arrangement under which a person is nominee or escrowee for another.
Trustee -- includes an original, additional, or successor trustee, whether or not appointed or confirmed by court.
Ward -- is as defined in § 62-5-101 (4) which states a "ward" is a person for whom a guardian has been appointed.
Will -- includes codicil and any testamentary instrument which merely appoints an executor or revokes or revises another will.