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\begin{center}
{\Large \bf QUINTESSENCE AS UNRECAPTURED MATTER FROM A UNIVERSE BOUNCE}\\
\vspace{1.0cm}
\baselineskip=14pt

\bf David E. Rosenberg\footnote{\copyright 2000, 
Electronic Mail: {\tt dxr1094@oak.njit.edu}}\\
{\em NJIT Physics Dept. Newark NJ 07102}\\
\end{center}
\baselineskip=24pt


\begin{quote}
\hspace*{2em} A core energy $\approx160$ MeV was required to propel the 
farthest galaxies at $0.5$c. The big bang baryon/photon ratio changes to
$\eta =87.6 \Omega_Bh^2$, comparable to a supernova.
Accretion photons emitted during the prior contraction supplied the CBR
planck spectrum at $2.73$ degrees. The expanding nonisotropic big bang 
recaptured this smooth planck spectrum, which contains evidence of prior 
emission. Collapse to infinite density states must be prevented by energy 
losses at supranuclear densities. A bounce occurring prior to a completed 
collapse phase is causing the negative vacuum energy 
$\Omega_M+\Omega_{\Lambda}=1.$
\vspace*{12pt}


\end{quote}

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\thesection{\centerline{\bf I. INTRODUCTION: \\ 
Limits on general relativity}}
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General relativity was discovered early this century and  
twenty four years after its introduction, it was found to predict 
black holes \re{Oppenheimer39a}. Relativity has been
extrapolated to where stars, galaxies and the whole universe
could be compressed into a space smaller than an atom. There is
not one shred of evidence that the universe started at Planck
densities $\rho=10^{93}g/cm^3$ and temperatures
$\rm{T}=10^{31}$ degrees. No high energy phenomena have been found from 
the first instant of creation. The nucleosynthesis of light
atomic nuclei ${}^4He$, ${}^2H$, and ${}^7Li$ took place around
densities of $10^5g/cm^3$ and temperatures of $\sim
10^{10}$ degrees, according to accepted models \reii{Peebles66}{Wagoner67}. 
These conditions are the most extreme that has been confirmed in
the big bang. Thus general relativity has been extrapolated eighty orders
of magnitude in density from points at which it has been
validated. The Einstein field equation is
\begin{equation}
R_{\mu\nu}-1/2Rg_{\mu\nu}\equiv G_{\mu\nu}=8\pi GT_{\mu\nu}+
\Lambda g_{\mu\nu}
\end{equation}
$G_{\mu\nu}$ is the Einstein tensor, $T_{\mu\nu}$ is the 
stress-energy tensor for all the matter and energy fields.
$\Lambda$ is a cosmological constant which has been recalled into use as the
universe has been found to be accelerating \reii{Perlmutter99}{Riess98}.
Based on the supernova Ia distance data, the geometry of the universe is
accelerating despite $\Omega_M = 0.3$. It seems that
the data to $z=1$ indicates that a negative energy, greater than all the
matter in the universe, is causing an acceleration
$\Omega_{\Lambda}\approx 0.7$.
There is no known reason a priori that vacuum energy should be on the order
of the total mass energy or that the two should total so close to the critical
density $\Omega_M + \Omega_{\Lambda} \approx 1.$ What could comprise 
negative energy is unknown. A re-evaluation of the entire inflation model is
thus in order.

For a homogenous, isotropic universe, the field equation has been simplified
to the Friedmann equation
\begin{equation}
\biggl ({\dot R\over R}\biggr)^2={{8\pi\rho G} \over 3 } - K {c^2\over R^2} +
{\Lambda \over 3} = H_o^2\biggl (\Omega_M + \Omega_{\Lambda} \biggr ) -
K{c^2\over R^2}\,,
\end{equation}    
where $H\equiv \dot R/R$ is the Hubble constant, which is
time dependent. G is the gravitational constant, $\rho$  is the
mass-energy density, p is the pressure and $R(t)$ is the scale
factor of the universe $\sim 10^{28}cm$ presently.
A black hole of mass M has a Schwarzschild radius $R_S$ and has a surface
area $4\pi R_S^2$. It has been shown that the universe must have started
as a black hole. A black hole having the mass of the universe would have
its gravitational radius $R_S \sim R(t)$. I will assume that the universe is
expanding into a region of space that was a prior black hole. 
The potential barrier enclosing the universe may be at
$R_S\approx 4\times 10^{28}cm.$
The geometry of the universe is still open as is it expands toward this 
barrier. Additional matter can be recaptured and change the internal geometry. 
There is evidence of an increase in photons in the CBR and matter as the
universe has expanded \reii{Skalsky00a}{Skalsky00b}.
An external view of the universe, assuming isotropy, can be constructed.
A static Schwarzschild metric, modified by vacuum energy, 
follows\re{Pathria72}.
\begin{equation}
ds^2={{dr^2}\over{1-{{2\mu}\over {r}}}-{{\Lambda r^2}\over 3}}+r^2(d\theta^2+
sin^2\theta d\phi^2)-\biggl (1-{{2\mu}\over {r}}-{{\Lambda r^2}\over 3}\biggr )
c^2dt^2\,.
\end{equation}
where $\mu=4\pi G\int_0^R{ } \rho(r)r^2dr$. Since $\Lambda>\Lambda_c$ and the
universe is accelerating, $R_{max}$ exists only if the potential barrier at
$R_S$ remains. $\int_0^R{ }\rho(r)dr$ will be additive only until $R=R_o$,
the current scale factor of the universe. Matter between $R_o$ and $R_S$ can
cause the negative vacuum energy in the right amount and time.

Classical general relativity is based on a perfect fluid in the
stress-energy tensor 
\begin{equation}
T_{\alpha\beta}=\rho \mu_\alpha \mu_\beta+p(g_{\alpha\beta} +
\mu_\alpha \mu_\beta) \,,
\end{equation}
which ignores viscosity, shearing forces and subatomic effects
including differences between individual baryons and bulk
baryons, ${n_B}\gg 10^3$. 
Since it has been validated up to nuclear densities in pulsars, 
changes in the stress-energy tensor $T_{\alpha\beta}$ 
at higher densities will be investigated.       


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\thesection{\centerline{\bf II. Theoretical changes for a bounce}}
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The Oppenheimer and Volkoff equations of
state \re{Oppenheimer39b} are used for neutron density matter
and neutron stars up to $3\times10^{14} gm/cm^3$, 
\begin{eqnarray}
dm\over {dr} &= {4\pi r^2\rho} \cr\cr 
dp\over dr &= -{(\rho+p)(m+(4\pi r^3p))}/
{r(r-m)},
\end{eqnarray}  
where m is the mass within a given radius r. Since these
equations result from the field equation, 
information about the density change with pressure 
is also necessary. Neutron stars theoretically
have masses up to $5M_\odot$. 
Single neutrons have a compression energy about 300 MeV to smash them
into quarks \re{Glendenning88}. Colliders start producing quark-gluon
plasma at energies over $2\times10^{12}$ degrees $\approx 184MeV$.
If matter in this temperature range has an abrupt (first order) 
transition, superheating can trigger explosions \re{Wilczek98}.
After all the space in the neutron is eliminated $\rho >10^{17}gm/cm^3$
with particle overlapping, the net quantum effect of further
collapse must be repulsion, neutron deformation and a reversible energy
sink. A reduction in kinetic energies including rotational and 
vibrational should result in a corresponding increase in potential 
energy. Since nuclear pressures can't halt a gravitational
collapse, sufficient energy loss at supranuclear densities must
result in a stable configuration prior to quark formation. An
inhomogeneous collapse must stop when the compression energy
losses of core neutrons at peak $\rho\sim 10^{18}-10^{19} g/cm^3$
exactly match the gravitational energy, as shown in figure 1. 


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\begin{center}
\thesection{\bf III. {Resulting changes in our understanding}}
\end{center}
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Prior to the big bang, core densities increased and the
energy sink losses rapidly overtook the collapse
energy by an overall mass-density $\rho\sim 10^{16}gm/cm^3$. 
If $0.3 \Omega_c$ of  the matter in the universe was in a spherical mass to
start, its radius was $\sim 10^{13}cm.$ About $0.7 \Omega_c$ was scattered 
at great distances, due to a previous expansion.  
As the density rose in the core of this mass, 
the field disappeared and the pressure $p=\rho/3
\to 0$ in the stress-energy tensor as well.    
With $T_{\mu\nu} =0$, the curvature tensor
$R_{\mu\nu} =0$ and the vacuum energy
$\Lambda< 0$, an open universe existed. 
No singularities ever existed since there were no infinities in
energy, density or time. Accretion and other photons from
previous universes were very red shifted
by release into volumes much greater than today, 
so that they played no role
during the open universe period. Neutron compression energy
supplied $\approx 160MeV\approx 1.85\times10^{12}$ degrees which
propelled the farthest galaxies $\sim 0.5c$. After the bounce,
the metric was nearly flat. 
There was no difference whether the early universe was closed 
or open \re{Misner73}. The extrinsic curvature $(6{\dot R}^2)/ R^2$ 
was much more important than the intrinsic
curvature $\pm 6/ R^2$ within any hyperspace of homogeneity,
since $\dot R^2$ was very large initially. The zones of influence
were too small to respond differently to negative or positive
spacetime curvatures.

The standard hot universe problems \re{Linde84}, can be
summarized and solved with the above correction.
The singularity problem follows from the scale factor of the
universe R(t) vanishes as $t\to 0$ and the energy density becomes
infinitely large. The inhomogeneity of matter with the energy
sink and red shifting of radiation prior to the big bang caused
the total energy-density $\to 0$.  

The flatness problem can be stated in several ways. The ratio
of the universe's mean mass and energy density to the cold Einstein-de
Sitter universe 
\begin{equation}
{\rho/\rho_c}= {3H^2}/{8\pi\rho G} \,. 
\end{equation}
The Friedmann-Robertson-Walker(FRW) equation implies that this
ratio, which was proportional to curvature, was 
$1\pm 10^{-60}$ at the Planck era. The kinetic energy $(\dot R/R)^2$ 
was equal to the gravitational mass-energy $8\pi\rho G/3$ ,
so that $K\approx 0$ in equation (1.2). 
Only a bounce mechanism by which the gravitational mass-energy 
was converted into kinetic energy could allow the universe to be
nearly flat without evidence of high energies such as monopoles.
$\sim 160$ MeV was sufficient to break the shell into billions of cold 
baryonic masses $\le 10^{16}M_\odot$. 
For mass M the gravitational radius is
$R_S= GM/c^2$. Then 
\begin{equation}
\rho= c^6/{G^3M^2} \,, 
\end{equation}
at black hole formation.  Thus primordial  holes could only be
formed from the expanding shell neutrons in masses $\ge 7M_\odot$
if $\rho_{max}\approx 10^{16}g/cm^3$. If
this density can not be exceeded, then smaller
black holes $<1M_\odot$ could not be formed, which would explain the
missing Hawking radiation \re{Halzen91}. It would also explain the finding of
six black holes all $\approx 7M_\odot$, none smaller \re{Bailyn97}.

The horizon problem has to do with areas in the initial
instant of creation that are too far from each other to have been
influenced by initial disturbances. A light pulse beginning at
t=0 will have traveled by time t, a physical distance 
\begin{equation}
l(t)= R(t){\int_0^tdt'}{ }R^{-1}(t')= 2t \,, 
\end{equation}
and this gives the physical horizon distance 
or Hubble radius dH. In a matter dominated universe before
vacuum energy $\Lambda$ became significant, 
\begin{equation}
dH\approx 2H_0^{-1}\Omega_0^{-1/2}(1+z)^{-3/2} \,, 
\end{equation}
where $\Omega_0= {\rho/ \rho_c}$ in the present universe. 
This distance is compared with the radius L(t) of the region 
at time t which evolves into our currently 
observed area of the universe $\approx 10^{10}years$. 
Using any model near Planck conditions, this ratio $l^3/L^3$ 
is going to be very small, about $10^{-83}$. 
The central mass has density initially  
$\sim 10^{16}g/cm^3$  rather than Planck densities of
$10^{93}g/cm^3$. By including the entire open universe, the 
horizon problem is eliminated. During the bounce of $0.3$ of the total matter, 
there was a nearly simultaneous release of the stored neutron compression 
energy. Since state data on bulk nucleons at supranuclear densities is 
lacking, a reduction equation for a static system is extrapolated 
for compression losses of $E_{sink}=exp(\rho /2\times10^{14})$ in the
energy term $T_{\hat o\hat o}$. 

The homogeneity and isotropy problems arise due to the
postulated start of the universe in such a state. The
distribution of galaxies and clusters are not quite random on large
scales. A compilation of 869 clusters has shown a quasi-regular pattern
with high density regions separated by voids at intervals
$\approx 120 Mpc$. \re{Einasto97}.  
It has long been assumed that galaxy formation, which started after 
the decoupling of matter and energy, grew by gravitational
amplification of small density fluctuations. 
With the Hubble space telescope, there is evidence that 
galaxies were assembled $z>4$ \re{Mo96}. Primordial galaxies,
composed of hot ${}^1H-{}^4He$ clouds orbiting the black hole
remnants of the cold shell, were present
prior to decoupling of matter and energy $z\approx 1100$. 
Evidence for this is found in the variation of primordial deuterium
by a factor of $10$. Large masses slow down the local expansion rate, 
allowing more time for deuterium to be converted to helium and correspond to 
a Jeans mass of $\sim 10^6 M_\odot$ \re{Copi96}. 
As the universe expanded and these shell remnants
separated, hydrogen was efficiently removed from intergalactic 
space down to the Gunn-Peterson ${}^1H$ limit, and attenuated the
CBR temperature gradients as follows.  Hot electrons upscattered
the redshifted photons emitted by orbiting hydrogen deeper in the
protogalactic wells. In primordial galaxies, free-free can dominate over 
Compton scattering between 90eV-1eV, lead to true thermalization and diminish
temperature gradients in the CBR.
Figure 2 diagrams the universe starting with the big bang expanding
in FRW geometry from an initial mass of radius $a(t)\sim 10^{13}cm.$ 
Distant matter which was outside the central mass (and expanding universe) 
is not included.  
Radiation energy $\rho_R\propto a^{-4}$ and $T\propto a^{-1}$. An increase
in R(t) from $10^{13}cm.$ to $10^{28}cm.$ today caused the
corresponding temperature of big bang photons to drop from
$1.85\times10^{12}$ Kelvins to $.00185$ Kelvin.
After the scale factor began to decrease from the maximum, galaxies began 
to merge R(t)$\sim 10^{25}cm.$ The smooth Planck spectrum at $2.73$ degrees
was released by accretion at $T\approx 2.73\times10^6\pm 16$ degrees 
when R(t)$\sim 10^{22}cm.$, as their nuclei merged.
Almost all the CBR photons were recaptured by the re-expanding universe and
were undisturbed by the big bang.
After the massive black hole lost energy by core compression,
open spacetime $T_{\mu\nu}=0$ propagated outward.

Figure 2 actually diagrams the thickness of $R_S$ as outside the radius
remained constant and the potential barrier remained in place.
Evidence for the existence of this barrier comes from the highest energy
gamma ray bursts. Particles coming into the expanding universe by crossing
this great potential are given $>10^{20}$eV energies 
without the energy cutoff at $5\times 10^{19}$eV due to CBR particle 
interaction. Other evidence for recapture of matter and photons comes from
the heavy metals found evenly dispersed in the intergalactic medium at high
red shift \re{Cowie98}. It is unnecessay to postulate isolated supernovas
at $z \approx 13-14$, prior to galaxy formation which occurred at $z\sim 5$
and well prior to star formation $z\sim 3$.

COBE satellite data showed a $\Delta T \approx 45$ 
microkelvins at CBR photon separations greater than 
$40^o$ and diminishing toward zero for lesser separations. These 
are plotted as $\Delta T^2$ versus angle of separation in figure 3, adopted
from Guth \re{Guth97}. The horizon distance at decoupling in degrees is
\begin{equation}
\theta(dH)=0.87^o\Omega_o^{1/2}(z_{dec}/1100)^{-1/2} 
\end{equation}
which is $\approx 0.8^o$ in the CBR today.
This temperature attenuation, which stretches over $40^o$ in 
the CBR, required extensive time for coupling of baryons and photons.
It began after accretion released the photons in the collapse phase, as
shown in figure 2. It lasted until the $K=-1$ curvature propagated far
enough to sufficiently decouple the photons and electrons R(t)
$\approx 10^{25}cm.$  
The photon number density $cm^{-3}$
\begin{equation}
n_\gamma =2.038\times10^{28}T_9^3 \,, 
\end{equation}
where $T_9$ is the temperature in units of $10^{9}$ degrees.
For the CBR in total there are $422 cm^{-3}$, over $99$ percent recaptured
during the expansion. For $T=.00185$ degree remnant of
the big bang, $n_\gamma =1.29\times10^{-7} cm^{-3}$. 
This changes its baryon/photon ratio to
\begin{equation}
\eta= 87.6{\Omega_B}h^2 \,, 
\end{equation}
where h is the Hubble constant in units of $100km.sec^{-1}Mpc^{-1}$. 
The explosion mechanism and $\eta$ are similar to that of a supernova.
The hot baryon to photon ratio must be multiplied by the cold
baryon factor CBF plus one to obtain the total baryon/photon ratio 
\begin{equation}
\eta_{total}=\eta_{hot}(CBF+1) \,. 
\end{equation}
An $\eta_{total} \approx 36$ will produce a flat universe if $h=0.66$ and
$\approx 11$ will match the current $\Omega_M =0.3$.
The nucleosynthesis program NUC123 of Larry Kawano was modified as follows. 
Cold baryons were calculated by multiplying the hot baryon
density thm(9) in subroutine therm by the cold baryon factor. 
This was added to the total energy density thm(10) and thus to 
the Hubble constant.  
Using double precision and cold baryons, neutrino degeneration and $\eta$ as
variables, it was found that $\eta=10^{-7}$, 
a cold baryon multiplier $10^8$ and an 
electron neutrino chemical potential $\xi_{\nu_e} =1.865$ 
gave a D or ${}^2H/H=1.64\times10^{-5}$ and a ${}^4He/H=.2352$. 
The deuterium fraction increased with increasing
cold baryons. The ${}^4He$ yields decreased with increasing
electron neutrino chemical potential by reducing the neutron 
to proton ratio at freeze out, as first noted \re{Wagoner67}.
Doubling the cold baryons gave a ${}^2H/H=2.89\times 10^{-5}$ without much
change to other yields. This supplies sufficient baryons for brown dwarfs for 
microlensing and other dark matter. 

Galaxy formation problems \re{Peebles90} are greatly
simplified. An explosive universe with galaxy formation will fit
the large scale galactic pattern \re{Weinberg89}. 
Although the Jeans mass is thought to be the point 
at which gravity overcomes pressure to form galaxies, massive
rotating primordial black holes are necessary for galactic
structure. In the Tully-Fisher relation 
\begin{equation}
V_c= 220(L/L_\star)^{.22} \,,
\end{equation}
and Faber-Jackson 
\begin{equation}
V_c= 220 (L/L_\star)^{.25} \,, 
\end{equation}
where $V_c$ is the circular velocity $km/sec$ and $L_\star$ 
is the characteristic galaxy luminosity. The former relation is
for velocities in the dark halo of spiral galaxies and the latter
for star velocity dispersion in central parts of elliptical
galaxies \re{Peebles93}. Rotational energy $E_{rot}$ is a function of
$MV_c^2$. Galactic brightness results from ${}^1H$ mass, $M_{galaxy}$. The
black hole capturing cross section 
\begin{equation}
\sigma_{capt.}=16\pi M^2/ \beta^2 \,, 
\end{equation}
where $\beta$ is the particle velocity relative to
light \re{Misner73}. Because of the ${}^1H$ capture by
primordial black holes, the brightness is proportional to the
central nuclear mass $M_{nucleus}^2$. With $M_{nucleus}^2V_c^4=$ constant, 
its square root is a constant related to the rotational 
energy imparted prior to the big bang. 
Thus the galactic mass and luminosity can be related
to the depth of the dark matter potential
well and asymptotic circular speed. Due to the capture mechanism
of ${}^1H$, the black hole nuclear mass $M_{nucleus}\propto M_{galaxy}$.
Galaxy formation never involved collapse dynamics
with its different post collapse densities, circular speeds and
disk asymmetries.   

The baryon asymmetry problem has been stated as to why there 
are many more baryons than antibaryons. Baryon-antibaryon pairs 
are only created from a vacuum at energies $>10^{13}$degrees, which
is higher than the $160MeV\approx 1.85\times10^{12}$degrees core
temperature. Extreme energy phenomena such as domain walls, monopoles,
gravitinos and symmetry breaking were not reached in the big bang.    

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\thesection{\centerline{\bf IV. A cyclical universe}}
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Although equation 1 is cyclical, it is valid only for a 
universe that is isotropic and homogeneous i.e. a perfect fluid.
After $R_{max}$ was reached in figure 2, the galaxies were blue shifted
as they reconverged. When R(t) was $10^6$ smaller than today, 
the proportionately higher CBR tore neutrons 
and protons from nuclei. In the center was a growing black hole 
resulting from merging galactic nuclei. Stars and galaxies were 
accreted onto this supermassive black hole in a massive thick
disk releasing the photons of the cosmic background. 
Once the mass of this black hole exceeded the size of an
average galactic nucleus $\sim 10^8M_\odot$, 
tidal forces were no longer capable of tearing a star apart
before it entered $R_S$ with relatively little radiative 
losses \re{Frank92}. The collapsing scale factor R(t) forced 
$\approx 0.3$ of the matter together prior to the re-expansion.  

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Although classical general relativity has been confirmed to one
part in $10^{12}$, it must break down prior to the infinite
densities of singularities. There is no reason why a small mass
$>7M_\odot$ can contract to a singularity while the mass of
universe explodes into the big bang. If a star surface lies
entirely inside the $R_S$, classical relativity concludes from
Kruskal-Szekeres diagrams that it must collapse to a singularity
or faster than the speed of light. Here coordinate reversal
occurs, $\partial/\partial r$ is timelike $(g_{rr}<0)$ and proper
time at the surface 
\begin{equation}
\tau= -\int^R\bigl [g_{rr}\bigr ]^{1/2}dr + constant \,. 
\end{equation}
In order to allow a big bang, a reduction in the 
stress-energy tensor must occur before enormous densities and
energies are reached inside $R_S$. As $T_{\mu\nu} \to 0$ 
quickly, the impetus for further collapse stops with energy loss.
After the limiting density is reached, there is re-reversal of the time
coordinate and no further reduction in size. 
The quantum requirement that $T_{\mu\nu} >0$, 
will not be violated as it will approach zero on the positive
side. A solution to the covariant perturbation problem for quantum
gravity would be as follows. The spacetime metric $g_{ab}$ is divided
into a flat Minkowski component $\beta_{ab}$ and its deviation 
$\gamma_{ab}$, where $(M,{}^og_{ab})$ is a solution to the field
equation. The field equation can be seen as an equation 
for a self interacting spin-2 field $\gamma_{ab}$ in Minkowski spacetime.
In the first order $\gamma_{ab}$ is a free spin-2 equation with much
gauge arbitrariness which can be expanded into a perturbation series 
for non-abelian gauge fields. Although this part is non-renormalizable, 
the energy sink correction eliminates this term at high energies leaving
the background metric $\beta_{ab}$ which satisfies causality conditions.
The quantum mechanism by which the energy sink suppresses vibratory and
other modes remains to be elucidated.
The problem of evaporation for black holes under a solar
mass due to quantum particle creation with violation of lepton
and baryon conservation is avoided. Naked and all other
singularities are mathematically eliminated. Black holes can
eventually influence their surroundings to achieve thermal
equilibrium. Supernovas $<7M_\odot$, when collapsing to  
the same limiting density, will bounce without black hole formation. 
A supranuclear equation of state based on actual data (which does not yet 
exist) and better nucleosynthesis modelling, taking into consideration
a gradient of temperature and all neutrino effects, will better 
determine $\eta$, hot and cold baryons and the bounce temperature.


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%%%%%%%%%%%%%%%   ACKNOWLEDGEMENTS  %%%%%%%%%%%%%%%
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\vspace{36pt}
Acknowledgements. This paper would not have been possible without
many helpful comments from J. Schiff, E. Kolb, D. Lindley, 
I. Klebanoff. It is an updated version of  Quintessence, Quantum Gravity 
and the Big Bang preprint astro-ph/9904320. 

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\begin{enumerate}

\item\label{Bailyn97} Bailyn, C.D., Jain, R.K. and Orosz, J.A.
preprint astro-ph/9708031

\item\label{Copi96} Copi, C.J., Olive, K.A. and Schramm, D.N.
preprint astro-ph/9606156

\item\label{Cowie98} Cowie, L.L. and Songalia, A. Nature 394,
44-46 (1998)

\item\label{Einasto97} Einasto, J. et al. Nature 385, 139-141
(1997) 

\item\label{Frank92} Frank, J., King, A. and Raine, D. Accretion
Power in Astrophysics (Cambridge University Press, Cambridge
England, 1992) 171-265

\item\label{Garnavich98} Garnavich, P.M. et al. Astrophys. J. 
Letts. 493, L53 (1998)

\item\label{Glendenning88} Glendenning, N.K. Phys. Rev. C 37,
2733-2742 (1988)

\item\label{Guth97} Guth, A.H. The Inflationary Universe (Addison-
Wesley Publishing Co., Reading Massachusetts, 1997) 239-243

\item\label{Halzen91} Halzen, F., Mac Gibbon J.H. and Weeks, T.C.
Nature 353, 807-814 (1991)

\item\label{Linde84} Linde, A.D. Rep. Prog. Phys. 47, 925-986 (1984)

\item\label{Misner73} Misner, C.W., Thorne, K.S. and Wheeler, J.A.
Gravitation (W.H. Freeman and Co., New York, 1973) 655-679,
703-816, 872-915

\item\label{Mo96} Mo, H.J., Fukugita, M. preprint astro-ph/9604034

\item\label{Oppenheimer39a} Oppenheimer, J.R. and Snyder, H. Phys.
Rev. 56, 455-459 (1939)

\item\label{Oppenheimer39b} Oppenheimer, J.R. and Volkoff, G.M.
Phys. Rev. 55, 374-381 (1939)

\item\label{Pathria72} Pathria, R.K. Nature 240, 298-299 (1972)

\item\label{Peebles66} Peebles, P.J.E. Astrophys. J. 146, 542-552
(1966)

\item\label{Peebles90} Peebles, P.J.E. and Silk, J. Nature 346,
233-239 (1990)

\item\label{Peebles93} Peebles, P.J.E. Principles of Physical
Cosmology (Princeton University Press, Princeton, N.J.,1993)
45-54, 527-564

\item\label{Perlmutter99} Perlmutter, S. et al. Astrophys. J. 517, 
565-586 (1999)

\item\label{Riess98} Riess, A. Astron. J. 116, 1009-1038 (1998)

\item\label{Skalsky00a} Skalsky, V. preprint astro-ph/0001251

\item\label{Skalsky00b} Skalsky, V. preprint astro-ph/0003319

\item\label{Wagoner67} Wagoner, R.V., Fowler, W.A. and Hoyle, F.
Astrophys. J. 148, 3-49 (1967)

\item\label{Weinberg89} Weinberg, D.H., Ostriker, J.P. and Dekel, A.
Astrophys. J. 336, 9-45 (1989)

\item\label{Wilczek98} Wilczek, F. Nature 391, 330-331 (1998)

\end{enumerate}
\end{document}


