\documentstyle[preprint,floats,prl,aps,epsf]{revtex}

\begin{document}

\preprint{\hbox {February 2003} }

\draft
\title{Stability Issues for $w < -1$ Dark Energy}
\author{\bf Paul H. Frampton
}
\address{
Department of Physics and Astronomy,
University of North Carolina, Chapel Hill, NC  27599.}
\maketitle
\date{\today}


\begin{abstract}
Global fits to precision cosmological data which
show that there is a dark energy component
comprising about 2/3 of the total cosmological energy also hint that a
dark energy with $w = P/\rho < -1$
is viable, even favored.
Here we discuss implications of such a surprising $w$,
including whether it jeopardizes vacuum stability.
It appears to be secure in microscopic processes,
but why bulk dark energy
has not decayed spontaneously
is mysterious.
\end{abstract}
\pacs{}

\newpage

\bigskip
\bigskip

\noindent {\it Introduction}.

\bigskip
\bigskip

\noindent The equation of state for the dark energy component
in cosmology has been the subject of much recent 
discussion
\cite{Melchiorri,Schuecker,Hannestad,Caldwell,Frampton,Bastero,Dicus,Ta,CHT}
Present data are consistent
with a constant  $w(Z) = -1$ corresponding to
a cosmological constant. But the data allow a present value
for
$w(Z=0)$ in the range
$- 1.62 < w(Z=0) < -0.74$ \cite{Melchiorri}.

If one assumes, more generally, that $w(Z)$ depends on $Z$
then the allowed range for $w(Z=0)$ is approximately the
same\cite{Ta}. In the present article we shall
forgo this greater generality as not relevant.
Instead, in the present article we address the question
of stability for a dark energy with constant 
$w(Z) < -1$.

\bigskip
\bigskip

\noindent {\it Behavior of $T_{\mu\nu}$ Under Boost}

\bigskip
\bigskip

\noindent Consider making a Lorentz boost along the
1-direction with velocity $V$ (put c = 1). Then
the stress-energy tensor which in the dark energy
rest frame has the form:
\begin{equation}
T_{\mu\nu} = \Lambda 
\left( 
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & w & 0 & 0 \\
0 & 0 & w & 0 \\
0 & 0 & 0 & w 
\end{array}
\right)
\label{restframe}
\end{equation}
is boosted to $T^{'}_{\mu\nu}$ given by

\begin{equation}
T^{'}_{\mu\nu} =
\Lambda
\left( 
\begin{array}{cccc}
1 & V & 0 & 0 \\
V & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 
\end{array}
\right)
\left( 
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & w & 0 & 0 \\
0 & 0 & w & 0 \\
0 & 0 & 0 & w 
\end{array}
\right)
\left( 
\begin{array}{cccc}
1 & V & 0 & 0 \\
V & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 
\end{array}
\right)
=
\Lambda
\left( 
\begin{array}{cccc}
1+V^2w & V(1+w) & 0 & 0 \\
V(1+w) & V^2+w & 0 & 0 \\
0 & 0 & w & 0 \\
0 & 0 & 0 & w 
\end{array}
\right)
\label{boost}
\end{equation}

\noindent We learn several things by studying Eq.(\ref{boost}).

First, consider the energy component $T_{00}^{'} = 1 + V^2w$.
Since $V < 1$ we see that for $w > -1$ this is positive $T_{00}^{'} > 0$
and the Weak Energy Condition (WEC) is respected\cite{hawking}.
For $w=-1$, $T_{00}^{'} \rightarrow 0$ as $V \rightarrow 1$ and
is still never negative. For $w< -1$, however, we see that 
$T_{00}^{'} < 0$ if $V^2 > -(1/w)$ and this violates the WEC
and is the first sign that the case $w < -1$ must be studied
with great care.

Looking at the pressure component $T_{11}^{'}$ we see the special
role of the case $w = -1$ because 
$w = T^{'}_{11}/T^{'}_{00}$ remains Lorentz invariant
as expected for a cosmological constant. Similarly
the off-diagonal components $T_{01}^{'}$
remain vanishing only in this case.

The main concern is the negativity of $T^{'}_{00} < 0$
which appears for $V^2 > -(1/w)$.

\bigskip
\bigskip

\noindent {\it Interpretation as Limiting Velocity}

\bigskip
\bigskip

\noindent One possibility is that it is impossible
for $V^2 > -(1/w)$. The highest velocities known are
those for the highest-energy cosmic
rays which are protons with energy $\sim 10^{20}eV$.
These have $\gamma = (1 - V^2)^{-1/2} \sim 10^{11}$
corresponding to $V \sim 1 - 10^{-22}$. This would imply
that:
\begin{equation}
w > -1 - 10^{-22}
\label{sillylimit}
\end{equation}
which is one possible conclusion.

\bigskip
\bigskip

\noindent {\it Interpretation as Vacuum Instability}.

\bigskip
\bigskip

\noindent But let us suppose, as hinted at by 
\cite{Melchiorri,Schuecker,Hannestad} 
that more precise cosmological data
reveals a dark matter which
violates Eq.(\ref{sillylimit}). Then, by boosting to an inertial
frame with $V^2 > -(1/w)$, one arrives at $T^{'}_{00} < 0$
and this would be a signal for vacuum instability\cite{PHF76}.
If the cosmological background
is a Friedmann-Robertson-Walker (FRW) metric the physics
is Lorentz invariant and so one should be able to see evidence
for the instability already in the preferred frame
where $T_{\mu\nu}$ is given by Eq.(\ref{restframe}).

This goes back to work in the 1960's and 1970's
where one compares the unstable vacuum to a superheated
liquid. As an example, at one atmospheric
pressure water can be heated carefully 
to above $100^0$ C without boiling.
The superheated water is metastable and attempts to nucleate
bubbles containing steam. However, there is an energy
balance for a three-dimensional bubble between the positive
surface energy $\sim R^2$ and
the negative latent heat energy of the 
interior $\sim R^3$ which
leads to a critical radius below which 
the bubble shrinks away and above which
the bubble expands and precipitates boiling\cite{Langer1,Langer2}.

For the vacuum the first idea in \cite{PHF76}
was to treat the spacetime vacuum as a 
four-dimensional material medium just like superheated water.
The second idea in the same paper 
was to notice that a hyperspherical bubble expanding at the speed of light
is the same to all inertial observers. This Lorentz invariance 
provided the mathematical  relationship
between the lifetime for unstable vacuum decay and
the critical radius of the four-dimensional 
bubble or instanton.

In the rest frame, the energy density is
\begin{equation}
T_{00} = \Lambda \sim (10^{-3} eV)^4 
\sim (10^{-3} eV)/ ({\rm mm})^{3} \sim (10^{-34} eV)/(100f)^3
\label{Lambda}
\end{equation}
since $10^{-3} eV \sim (1 {\rm mm})^{-1}$. In the last expression
of Eq. (\ref{Lambda}) $10^{-34} eV \sim H_0$
is the present Hubble constant which numerically is the dark energy
contained in about a million cubic fermis; these alternatives will be used nearer 
the end of this article.

In order to make an estimate of the 
dark energy decay lifetime
in the absence of a known potential,
we can proceed by assuming
it is the same Lorentz invariant 
process of a hyperspherical bubble
expanding at the speed of light, the same for all
inertial observers.

Let the radius of this hypersphere be R, its energy density be $\epsilon$
and its surface tension be $S_1$. Then according to \cite{PHF76}
the relevant instanton action is
\begin{equation}
A = -\frac{1}{2} \pi^2 R^4 \epsilon + 2 \pi^2 R^3 S_1
\label{action}
\end{equation}
where $\epsilon$ and $S_1$ are the volume and
surface energy densities, respectively.

The stationary value of this action is
\begin{equation}
A_m = \frac{27}{2} \pi^2 S_1^4 /\epsilon^3
\label{stationaryA}
\end{equation}
corresponding to the critical radius
\begin{equation}
R_m = 3S_1 / \epsilon
\label{Rcritical}
\end{equation}
We shall assume 
that the wall thickness is negligble compared to
the bubble radius.
The number of vacuum nucleations in the
past lightcone is estimated
as
\begin{equation}
N = (V_u \Delta^4) exp ( - A_m)
\label{nucleations}
\end{equation}
where $V_u$ is the 4-volume of the past and
$\Delta$ is the mass scale relevant to the
problem.

This vacuum decay picture led
to the proposals of inflation\cite{Guth},
for solving the horizon, flatness and monopole problems 
(only the horizon problem was generally known at
the time of \cite{PHF76}).
None of that work addressed why the true vacuum has zero energy.
Now that the observed vacuum has non-zero energy
density
$+ \epsilon \sim (10^{-3} eV)^4$
we may interpret it as a ``false vacuum" lying
above the ``true vacuum" with $\epsilon = 0$.

In order to use the full power 
of Eqs. ( \ref{action}, \ref{stationaryA}, \ref{Rcritical},
\ref{nucleations})
taken from \cite{PHF76,PHF77} we need
to estimate the three mass-dimension parameters
$\epsilon^{1/4}, S_1^{1/3}$ and $\Delta$ therein.
Let us discuss these three scales in turn.

The easiest of the three to select is $\epsilon$. If we imagine a
tunneling through a barrier between a false vacuum
with energy density $\epsilon$ to a true vacuum at energy density zero
then the energy density inside the bubble will
be $\epsilon = \Lambda = (10^{-3} eV)^4$. No other
choice is reasonable.

Next we discuss the typical mass scale $\Delta$ in the prefactor
of Eq. (\ref{nucleations}). Here we argue that the value of $\Delta$
does not matter. Suppose we take the largest imaginable scale, the Planck scale, 
$M_{Planck}$. Then the prefactor has the value $\sim 10^{250} \sim e^{575}$.
But even this is easily compensated by the factor $exp ( - A_m )$.
To ensure $N < 1$ one needs $ A_m > 575$ which means:
\begin{equation}
\left( \frac{S_1^{1/3}}{\epsilon^{1/4}} 
\right)^{12} > \frac{575}{\frac{27}{2} \pi^2} \sim 4.3
\label{inequality}
\end{equation}
Given that $(4.3)^{1/12} \sim 1.13$ this requires only that
$(S_1)^{1/3} > (\epsilon)^{1/4} \sim 10^{-3} eV$.
This will ensure $N<1$ for any smaller $\Delta$ and
thus the prefactor, and the scale $\Delta$,
does not strongly influence
vacuum stability just as is the case in field theory.

The most difficult scale to estimate is the value of $S_1$. 
Given that the dark energy
has not already ``boiled" would require that $S_1 \gg (10^{-3} eV)^3$.
But why this value of $S_1$? This question could
be answered given a potential barrier to
characterize the transition between dark energy and
true vacuum. Here the potential is unknown but
we would expect dimensionally that
$S_1 \sim (10^{-3} eV)^3$.
Given the inequality on 
$S_1$ we know that the critical radius of a nucleation
bubble would be macroscopic since, from Eq.(\ref{Rcritical}),
$R_m > 1$ mm.
This means that any smaller quantity of dark energy,
with total energy $E < 10^{-3} eV$,
will be of a subcritical size and hence unable to
make the transition to the real vacuum. The observed
stability of the dark energy
suggests $N < 1$ and hence an upper limit
to the dynamical length size ($L < 1{\rm mm}$).

Let the quantity of dark energy which would
decay, were Eq.(\ref{inequality}) violated, be $E$ in eV units. 
Assuming that the dynamical mass scales which generate 
$E$ are the quantities $M_{Planck}$ and $H_0$ we may assume 
from dimensional analysis that for some $p$
\begin{equation}
E = (H_0^p M_{Planck}^{1-p})
\label{Evalue}
\end{equation}
If we write $\Lambda = (H_o M_{Planck})^2 = E/L^3$ then
\begin{equation}
L = (H_0^{p-2} M_{Planck}^{-1-p})^{1/3}
\label{Lvalue}
\end{equation}
and we now express theoretical prejudices about $p$.

The first observation about $p$ is that it is inconceivable that it fall
outside of the range
$-1 \leq p \leq 2$ because, if $p=-1$, $L$ is the radius of the visible universe
while, if $p=2$, $L$ is the Planck length.
Our previous discussion of dark energy stability suggests
that $p \geq 1/2$ in order that $L \leq 1$ mm, so
eliminating $-1 \leq p < 1/2$. 

This brings us to the question of whether such
decay can be initiated in {\it e.g.} a
high energy particle collision.
This was first raised in \cite{PHF76}
and revisited for cosmic-ray collisions
in \cite{Hut}. That was in the context of
the standard-model Higgs vacuum and the
conclusion is that high-energy colliders
are safe at all present and planned foreseeable energies because
much more severe conditions have already
occurred (without disaster) in cosmic-ray
collisions within our galaxy.
More recently, this issue has been addressed
in connection with fears
that the Relativistic Heavy Ion Collider
(RHIC) might initiate a diastrous 
transition
but according to careful analysis\cite{JBWS,AdR}
there was no such danger. 

Unimodular gravity \cite{Sorkin,NV} can be interpreted
to suggest $p = 1/2$ \cite{YJN} in Eq.(\ref{Evalue}).
The arguments in
\cite{Sundrum} compellingly suggest similarly the scale corresponding to
$p = 1/2$ in Eq.(\ref{Evalue}) and (\ref{Lvalue})
{\it i.e.} $E \sim 10^{-3} eV$
which is the dark energy contained in a volume $\sim (1mm)^3$.
The discussion of \cite{Sundrum} suggests that the graviton
could be composite with a scale in a range around $10^{-3} eV$ and that the
interactions with standard model
particles are not point-like but via
``stringy'' halos.
This could cut-off in the ultra-violet the contributions of the
QCD and weak scales in the effective
gravity theory, and explain why the corresponding much-too-large
vacuum energies do
not show up in the vacuum
energy appearing on the Friedmann equations.
The graviton-compositeness scale
cannot be much larger than
$10^{-3} eV$ otherwise the cosmological
term would be too large; it
cannot be much smaller, otherwise deviations from
Newton's inverse square law would already
have been detected in terrestrial
experiments such as \cite{Adelberger}.

The case $p = 1/2$ is by far the most plausible since any
$p > 1/2$ invokes a new mass scale.
For $p = 1/2$, stability in the sense of Eq.(\ref{nucleations})
requires, from Eq.(\ref{Rcritical}), that $R_m \gg 1mm$ which 
implies $S_1 \gg (10^{-3} eV)^3$ in order to  
ensure, for stability, the absence
of nucleations $N \ll 1$.
But qualitative arguments
would instead suggest, in the absence of another
mass scale, that $S_1 \sim (10^{-3} eV)^3$ in which case
the observed stability of the bulk dark energy 
is rendered a mystery which
leads to our {\it dramatic} conclusion about $w < -1$.
The type of dark energy instability discussed
in this paper merits consideration.

As a first remark, since the critical radius $R_m$
for nucleation is macroscopic, it appears that
the instability cannot be triggered by any 
microscopic process.
While it may be comforting to know that the 
dark energy is not such a doomsday phenomenon, it
also implies at the same time the dreadful conclusion 
that dark energy may have no 
microscopic effect.
If any such microscopic effect in a terrestrial 
experiment could be 
found, it would be crucial in 
investigating the dark energy phenomenon.

In closing one may speculate 
how such stability arguments may evolve in the future.
As prognostications, we may expect most conservatively that
the value $w = -1$ will eventually be established empirically in which case 
both quintessence and the ``phantom menace'', as well
as the stability arguments presented above,
will all be irrelevant. 
In that case, indeed for any $w$, we may still hope that dark energy will
provide the first connection
between string theory and the real
world as in {\it e.g.} \cite{Bastero}.
If more precise data do establish $w < -1$,
as in the ``phantom menace'' scenario, the 
stability arguments presented here seem
so serious that,
since the data suggest that the dark energy
has existed for gigayears
and the theoretical arguments suggest it decayed long ago,
one is led to question
the validity of the Friedmann equations or, in other words,
of classical general relativity.

\bigskip
\bigskip
\bigskip


\noindent {\it Acknowledgements}

\bigskip

This work
is supported in part by the
US Department of Energy under
Grant No. DE-FG02-97ER-41036.
 
\newpage




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