%Paper: 
%From: leszek@leszek.physics.lsa.umich.edu (Leszek Roszkowski)
%Date: Mon, 25 Jan 93 12:47:26 EST
%Date (revised): Mon, 25 Jan 93 13:29:49 EST

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\begin{titlepage}
\begin{flushright} RAL-93-003\\UM-TH-33/92\\ December 1992\\
\\
\end{flushright}
\vskip 2cm
\begin{center}
{\bf\large Implications for Minimal Supersymmetry from
Grand}	%\\
\vskip 4pt
{\bf\large
Unification and the
Neutralino Relic Abundance}
\vskip 1cm
{\bf  R. G. Roberts\\}
\vskip 2pt
{\it
Rutherford Appleton Laboratory,\\
Chilton, Didcot, Oxon,
OX11 OQX, England}\\
\vskip .25cm
{\it and}
\vskip .25cm
{\bf Leszek Roszkowski\\}
\vskip 2pt
{\it Randall Physics Laboratory, University of Michigan,\\ Ann Arbor,
MI 48190, USA\\
leszek@leszek.physics.lsa.umich.edu}\\
\end{center}
\vskip .5cm
\begin{abstract}
We examine various predictions of the minimal supersymmetric standard
model coupled to minimal supergravity.  The
model is characterised by a small set of parameters at the
unification scale.
The supersymmetric particle spectrum at low energy and the
spontaneous
breaking of the standard model itself are then generated radiatively.
The previously considered
predictions of the model now include the neutralino relic density
which in turn
provides bounds on the scale parameters. We find a remarkable
consistency among several different constraints which imply all
supersymmetric particle masses preferably within the reach of future
supercolliders (LHC and SSC). The requirement that the neutralino be
the dominant component
of (dark) matter in the flat Universe provides a {\em lower} bound
on the spectrum of supersymmetric particles
beyond the reach of LEP, and most likely
also the Tevatron and LEP 200.


\end{abstract}
\end{titlepage}
\setcounter{page}{2}
\section{Introduction}

The minimal extension of the standard model~\cite{susyrev} which
corresponds to a
softly-broken supersymmetric $SU(3)\times SU(2)_L \times U(1)_Y$ at
the scale
$\mgut$ where the gauge couplings unify (as recently confirmed
by LEP~\cite{amaldi}) provides a very attractive
and
economic description of physics beyond the standard
model.  It is
possible to
specify a small number of parameters at the unification scale and the
low
energy effective theory is then determined simply by the radiative
corrections.
In particular the spontaneous breaking of electroweak symmetry is
radiately
generated due to the presence of supersymmetry soft-breaking terms
through the mass squared of one of the two Higgs doublets being
driven negative at the scale $Q\simeq{\cal O}(\mz)$ by the Yukawa top
quark
coupling~\cite{{gsymbreak}}.
In terms of the starting parameters at the GUT scale a detailed
spectrum of
the supersymmetric (SUSY) states is completely determined.
Even in the simplest SUSY scenario one meets considerable uncertainty
related to the presence of both the superheavy states around the GUT
scale and, more importantly, new supersymmetric states above
$\mz$~\cite{bh,meshkov,rr}.
Clearly the corresponding threshold corrections around $\mgut$ depend
on which unified
group or superstring scenario the minimal SUSY model is embedded
into. This inherent uncertainty would weaken the predictive
power
of the theory and, as in a previous paper~\cite{rr}, we perform a
minimal analysis where such corrections are ignored but corrections
from supersymmetric states above $\mz$ are treated with particular
care~\cite{rr}. Similarly,
the important constraint coming from the limits on the proton
decay~\cite{protondecay}
depends on
the choice of a specific GUT model and will not be discussed
here.


In this letter, we extend the previous analysis~\cite{rr} to include
the
predictions for the relic abundance of the lightest neutralino $\chi$
which
is typically the lightest supersymmetric particle (LSP) of the model.
The neutralino LSP has
long been identified~\cite{ehnos} as one of the leading candidates
for dark matter
in the Universe~\cite{dmrev,kt}. It is neutral, weakly interacting,
stable (if $R$-parity is valid) particle and its relic density is
typically consistent with present cosmological expectations.
We examine the predictions for $\chi$  from the minimal
supersymmetric standard model (MSSM) and compute the annihilation
cross
sections which requires the detailed knowledge of the whole SUSY
spectrum.
Consequently we can relate values of the neutralino relic abundance
to values
of the parameters $\mhalf, \mzero, \muzero$ --- the common gaugino
mass, the
common scalar mass and the higgsino mass at $\mgut$.
The lower limit on the age of the Universe provides an upper bound on
the
relic abundance of matter, and in particular of dark matter which is
believed to be a dominant mass component of the Universe.
We
can therefore use the dark matter abundance constraint to
derive bounds on the ranges of $\mhalf, \mzero, \muzero$ and in turn
get
constraints on the masses of all the SUSY particles.

In fact we can combine the dark matter constraint with others which
are
either
phenomenological (values of $\alpha_s (\mz), \mtop, \mbot)$ or
theoretical
(avoiding mass hierarchy problem) and examine the consistency of
trying to
satisfy several of these constraints simultaneously.  We conclude
that one can
indeed achieve such consistency quite naturally.
More interestingly, we find that this happens for
the ranges of the fundamental
parameters $\mhalf$, $\mzero$, $\muzero$, and thus also
masses of the supersymmetric particles, all preferably
within the few hundred GeV mass range and thus
{\em well within the reach} of the SSC and the LHC but typically
{\em above the reach}  of LEP, the Tevatron, and LEP 200. The lower
limit on supersymmetric particle masses comes from the dark matter
constraint as will be discussed in section 3.

The LSP for which we find sufficiently large values of the relic
abundance
to explain at least DM in the galactic halos ($\abund\gsim 0.025$)
invariably comes out to be almost
gaugino-like (bino-like) consistent with the
conclusions of some previous analyses~\cite{chiasdm,sugradm,dn}. It
was first noticed in
Ref.~\cite{chiasdm}
that a higgsino-like LSP is somewhat disfavoured as it corresponds to
a
high scale of supersymmetry breaking, typically exceeding 1 TeV, and
thus a gaugino-like LSP was selected as a unique candidate for DM.
More
recently, it has been
shown~\cite{dn,japan}
that for the higgsino-like neutralinos additional effects
(co-annihilation with the next-to-lightest
neutralino and the lightest chargino, see sect.~3) have a dramatic
effect of reducing the LSP relic
abundance below any interesting level. Here we find that
higgsino-like
LSPs are also largely excluded by the
current lower bound on the mass of the top quark.

Overall, the LSP relic abundance
constraint, combined with
the other constraints narrows down the allowed ranges of $\mhalf,
\mzero,
\muzero$ considerably. We find that the region $\mhalf\gg\mzero$ is
excluded by the lower bound on the top mass,
while in the region $\mhalf\ll\mzero$ the LSP relic abundance is too
large ($\abund>1$). Furthermore, the requirement that the LSP provide
enough
missing mass in the flat ($\Omega=1$) Universe can be fulfilled only
in
a relatively narrow band of comparable values of $\mhalf$ and
$\mzero$ and for $1<\muzero/\mzero\lsim {\rm a~few}$.

In the next section we briefly review and update the procedure
used in deriving the low-energy spectrum from a limited number of
basic parameters at the GUT scale. In section 3 we calculate the
neutralino relic density and compare it with other constraints on
the parameter space. We conclude with final remarks in section~4.


\section{Solutions of the MSSM}

We consider the MSSM in the context of a unified theory.  At
the
compactification scale $\mgut$ where the three couplings of
$SU(3),SU(2),U(1)$
have a common value $\alpha_X$ the SUSY parameter space is
characterised by
the common values of the gaugino masses $\mhalf$, the common value of
the soft mass terms of the squarks, sleptons and Higgs bosons
$\mzero$,
and by $\muzero$, the mass parameter of the Higgs/higgsino bilinear
term
in the superpotential. (The suffix 0
denotes values at $\mgut$.) In addition there are two parameters
characterising the soft terms proportional to the superpotential
terms: $B_0$ in the bilinear term $B_0\muzero$, and a common
trilinear soft parameter $A_0$ which multiplies the Yukawa terms.
Also one should
include at least the Yukawa couplings $h_{t0}, h_{b0}, h_{\tau 0}$ at
$\mgut$ to consider as parameters in principle.


However we can reduce this apparently
unmanageable host of parameters
down to a manageable set as follows.
Firstly the coefficients $A_0$ and $B_0$ are set to zero,
in the spirit of string-derived versions of the model~\cite{rr}.
Below the
scale $\mgut$,
$B$ grows to a finite positive value and generally reaches a maximum
and
may even decrease to negative values.  The values of $\mgut$ and
$\alpha_X$
are determined by the unification of the gauge couplings.  Their
precise values
for each solution are computed by an iterative procedure discussed in
Ref.~\cite{rr} since the running of the gauge couplings depends on
knowing the individual
SUSY thresholds which in turn depend on all the parameters including
$\mgut$
and $\alpha_X$ themselves.  This procedure requires the measured
values of
$\alpha_1 (\mz), \alpha_2 (\mz)$ from LEP but the value of $\alpha_3
(\mz)$ must be adjusted to achieve the required unification for each
solution.
Another adjustment is to choose $h_{t0}$ such that the running Higgs
mass
squared $m^2_2(Q)$ takes on the precise value (negative and
${\cal O}(\mz^2$)) at
$Q=\mz$ needed to give the required spontaneous breaking of
electroweak
symmetry, \ie,
\be
(m^2_1-m^2_2) + (m^2_1+m^2_2) \cos 2\beta = - \mz^2 \cos 2\beta
\label{gbcondition}
\ee
where $m_1,m_2$ are the running masses of the Higgs doublets
coupling to
down- and up-type quarks respectively.  Here the ratio of the Higgs
{\it v.e.v.}s
$v_2/v_1 = tan\beta = \cot\theta$ with $\beta$ related to $\mu$ and
$B$ by
$\sin 2\beta =2B\mu /(m^2_1+m^2_2)$.  The running of $m^2_2$ and
therefore the satisfying of eq.~(\ref{gbcondition}) is controlled by
the value of $h_{t0}$.
Actually the other significant Yukawa couplings $h_b, h_\tau$ should
be included in this running
of $m^2_2$ but in order to achieve eq.~(\ref{gbcondition}) in a
controllable way we drop them
which is justified as long as $\tan\beta$ is not too large
($\tan\beta\ll\mtop/\mbot$).

Thus each solution is specified by the values of the three parameters
$\mhalf$, $\mzero$, $\muzero$.  Each solution then provides at low
energies
specific
values for the quantities $\alpha_s (\mz)$, $\tan\beta,\mu$, gaugino
masses $M_1,M_2,M_3$, squark masses, slepton masses, Higgs masses,
Higgsino masses and top quark mass $\mtop =(\sqrt{2}h_t
\mw/g)\sin\beta$.
Relaxing the constraints $A_0=0, B_0=0$ affects the resulting value
of
$\tan\beta$ mostly --- see the analysis of Ref.~\cite{rr}, and so
quantities which
depend sensitively on $\tan\beta$ at low energies are, in principle,
less
precise, in our procedure.  From Ref.~\cite{rr} we see that, in
general, $\tan\beta >2$ even when $A_{0}, B_0$ are allowed to vary
within values of ${\cal O}(m_0$).

Another quantity associated with each solution is the ratio
$\mbot/m_\tau$,
assuming that this ratio is unity at $\mgut$.  Thus we include also
$h_{b0}=h_{\tau 0}$
as another parameter in the running of the Yukawa couplings, and
obtain
a specific value for $\mbot$ for each solution.  Apart from the above
phenomenological constraints on the solutions we have the standard
constraints that the Higgs potential be bounded, \ie,
\be
|\sin 2\beta|<1,\;\;
m^2_1m^2_2<\mu^2 B^2
\ee
and that all the physical mass squared be
positive.

The strongest constraint for insisting that the SUSY spectrum is
relatively
light comes from the `naturalness' argument~\cite{rr,bg} which
regards
the need to tune the value
of $h_{t0}$ to a very high precision in order for $m^2_2$ to take the
exact
value given by eq.~(\ref{gbcondition}) at the scale $Q=\mz$.  A
measure
of this `fine
tuning'
problem is the fine tuning constant $c$ defined by~\cite{rr}
\be
c = \frac{\delta \mw^2}{\mw^2}/\frac{\delta h^2_t}{h^2_t}
\label{cdef}
\ee
so that absence of fine tuning means $c \approx 1$. Approximately
we have~\cite{rr}
\be
c \approx \frac{m_0^2 + \mu_0^2 + k \mhalf^2}{m_Z^2}
\label{finetuning}
\ee
A reasonable limit to the degree
of precision needed would be $c \lsim {\cal O} (10)$ and
consequently the
typical
SUSY mass cannot be many times greater than $\mz$.

We illustrate the various constraints by showing the values of
$\mtop$ and $\mbot$ in Fig.~\ref{figone}a and
$\alpha_s(\mz)$  and $c$ in Fig.~\ref{figone}b as a function of
$\mhalf$ and $\mzero$ for a fixed ratio $\muzero/\mzero=2$.
The variation with $\muzero/\mzero$ will be discussed later.
The regions marked CDF and LEP are excluded by the CDF searches for
the top ($\mtop\gsim 91$ GeV) and the LEP searches for charginos
($\mcharone\gsim 46$ GeV), respectively.
We see from Fig.~\ref{figone}a  that the current `experimental' value
for $\mbot$ (in the $\overline{\rm MS}$ scheme),
$\mbot(2\mbot)= 4.25\pm 0.1$ GeV~\cite{bottom}, implies a rather
heavy
top
quark ($\mtop\gsim 150$ GeV) for the values of the input
parameters
$\mhalf,\mzero$ and $\muzero$ roughly within the 1 TeV limit. On the
other hand, beyond that range the resulting value of $\mbot$ is
consistent with $\mtop\lsim 150$ GeV. Larger values of the input
mass
parameters are, however, disfavoured by the fine-tuning constraint
and
the current bounds on $\alpha_s=0.122\pm0.010$ (based on analysis of
jets
at LEP)~\cite{altar}
as we can see from Fig.~\ref{figone}b.
We also note that the uncertainty on $\tan \beta$ arising from
allowing
$A_0$ and $B_0$ to be non-zero (discussed above) would imply that
$\mtop$
could be smaller by a further 10\%.

One
can see immediately that demanding $c\lsim {\cal O} (10)$ forces
one
to consider
only values of $\mhalf,\mzero$ up to a few hundred GeV.  This was the
conclusion of the previous analysis~\cite{rr}.  Thus unification of
the couplings
demands a value of $\alpha_s(\mz)$ close to the values extracted from
jet
analyses at LEP.

To summarise so far, the solutions obtained for the MSSM with the
inclusion of
electroweak symmetry allow a fairly restricted region of the
parameters
$\mhalf,\mzero,\muzero$ which is consistent with all the above
constraints, \ie,
$\mhalf,\mzero\lsim$ 200 GeV, $\muzero\lsim$ 400 GeV.
We will comment on the restrictions on the ratio $\muzero/\mzero$
later.

\section{The Neutralino Relic Abundance}

The knowledge of the whole mass spectrum of both the ordinary and
supersymmetric
particles allows one to reliably compute the relic abundance of
the lightest supersymmetric particle (LSP)
as a candidate for the dark matter in the Universe.

At the outset we note that, in the parameter space not already
excluded by LEP and CDF, we find that it is the lightest of the four
neutralinos that is {\em always} the LSP.
Another potential candidate for the LSP, the sneutrino, has been now
constrained by LEP to be heavier than about 42 GeV
and, if it were the LSP, its contribution to the relic abundance
would
be of the order of $10^{-4}$, and thus uninterestingly small. In the
analysis presented here, the sneutrino is typically significantly
heavier than the lightest neutralino. Typically, it is not even the
lightest sfermion.

The actual procedure of calculating the relic abundance has been
adequately described in the literature and will not repeated here.
We use the technique developed
in Ref.~\cite{swo} which allows for a reliable (except near poles and
thresholds) computation of the thermally
averaged annihilation cross section in the non-relativistic limit and
integrating the Boltzmann equation.

In the early Universe the LSP pair-annihilated into ordinary matter
with
total mass not exceeding $2\mchi$. In calculating the LSP relic
density
one needs to include all possible final states. Lighter $\chi$s
annihilate only
(except for rare radiative processes) into
pairs of ordinary fermions via the exchange of the $Z$ and the Higgs
bosons, and the respective sfermions. As $\mchi$ grows new final
states
open up: pairs of Higgs bosons, gauge and Higgs bosons, $ZZ$ and
$WW$,
and $t\bar t$. We include all of them in our analysis.

Generally one considers $\abund>1$ as incompatible with the
assessed lower bound of about 10 Gyrs on the age of the Universe or,
in
more popular terms, as corresponding to too much mass in the
Universe~\cite{kt}. Many astrophysicists strongly favour the value
$\Omega=1$ (or very close to one), corresponding to the flat
Universe, either because of cosmic inflation
or for aesthetical reasons. Moreover, there is growing evidence that,
on a global scale, the mass density indeed approaches the critical
density, as well as that most of the matter in the Universe is
non-shining and non-baryonic~\cite{kt}.
If one assumes that the LSP is the dominant component of dark matter
in the flat ($\Omega=1$) Universe then one typically expects
\be
0.25\lsim\abund\lsim0.5,
\label{flatuniv}
\ee
where the biggest uncertainty lies in our lack of knowledge of the
Hubble parameter $h_0$ to better than a factor of two. As we will
see shortly, varying somewhat the bounds in
eq.~(\ref{flatuniv}) will not significantly alter our conclusions.

We present in Fig.~\ref{figone}c the relic abundance of the LSP
and compare it with the other results shown before in
Figs.~\ref{figone}a and~\ref{figone}b. Several features can
be immediately noticed.

Firstly, most of the region corresponding to larger values of
$\mzero$ (roughly $\mzero\gsim$ 500 GeV)
is cosmologically excluded
as it corresponds to $\abund>1$. The relic abundance generally
decreases with decreasing $\mzero$ reaching very low values of
$\abund$ (0.025, or less) for $\mzero$ roughly below 200 GeV,
especially for $\mhalf>\mzero$.
It is worth noting that the region favoured by cosmology,
eq.~(\ref{flatuniv}),
takes a shape of a relatively narrow band running roughly parallel to
the border of the area excluded  by $\abund>1$. The contour
$\abund=0.1$ shows how quickly $\abund$ decreases with decreasing
$\mzero$ but also limits from below the region where the LSP relic
abundance is reasonably large.

It is interesting to see what mass and compositions of the LSP
correspond to
its relic abundance favoured by cosmology. We remind the reader that,
in minimal supersymmetry, the lightest neutralino
and its three heavier partners $\chi^0_i$ ($i=1,...,4$)
are the physical (mass) superpositions of
higgsinos $\hinob$ and $\hinot$, the fermionic partners of the
neutral Higgs bosons, and of two gauginos $\bino$ and $\wino$,
the fermionic partners of the neutral gauge bosons
\be
\chi\equiv
\chi_1^0=N_{11}\widetilde W^3+N_{12}\widetilde B+N_{13}\widetilde
 H_1^0+N_{14}\widetilde H_2^0.
\label{chilsp}
\ee
In distinguishing the gaugino-like and higgsino-like regions it
is convenient to use the gaugino purity $p=N_{11}^2+N_{12}^2$. In
particular, the LSP is almost a pure
bino where $p_{bino}\equiv N_{12}^2$
is close to one.
In Fig.~\ref{figone}d we show the bino purity of the LSP. (The
gaugino
purity is almost identical.) Remarkably, we find that the band
favoured by cosmology corresponds to the LSP being almost a pure bino
($\sim95\%$) up to very large values of $\mhalf$. We also find that
that higgsino-like LSPs are incidentally almost entirely excluded by
the lower
bound on the top quark of 91 GeV. (The contour of equal gaugino and
higgsino contributions almost coincides with the contour $\mtop=91$
GeV.) It was also noticed in Ref.~\cite{dn} that for a heavy top
constraints from radiative gauge symmetry breaking exclude
higgsino-like LSPs.
(With the expectation for $\mtop$ to be actually much heavier than 91
GeV a larger cosmologically uninteresting region is likely to be
ruled out.)
The LSP mass contours are almost vertical in the gaugino region with
$\mchi$ growing with $\mhalf$, and
almost horizontal in the higgsino region with $\mchi$ increasing with
$\mzero$. Again, the lines meet in the narrow sub-diagonal region
where
the LSP is both a  gaugino and a higgsino.

Since higgsino-like LSPs in our analysis not only give very little DM
but also are practically excluded by the CDF top searches, we need
not
worry about the additional effect of the higgssino-like LSP
`co-annihilation'~\cite{gs} with the next-to-lightest neutralino and
the lightest chargino which has been recently shown to significantly
reduce the LSP relic density~\cite{dn,japan}.
We have explicitly verified that {\it all} solutions for
which
co-annihilation of the LSP with $\chi_2$ and $\chi^\pm_1$ is
important
lie in the region excluded by $\mtop\geq 91$~GeV. Thus neglecting the
effects of co-annihilation is justified.

The LSP relic abundance in the allowed region is mostly dominated
by its annihilation into
fermionic final states, although in a few cases the Higgs final
states
contributed comparably. We thus do not expect that the radiative
corrections to the Higgs masses due to the heavy top~\cite{radcorrs}
would noticeably modify our results~\cite{sugradm}. We also found
that the lightest sfermion is
either $\stopr$ or $\slr$, in agreement with Ref.~\cite{dn}, except
in the (mostly excluded by LEP) region of small $\mzero$ and $\mhalf$
where it is the sneutrino.

We now pass to combine the band favoured by cosmology with
the mass contours of the top and the bottom quarks. This is shown
in Fig.~\ref{figtwo}. We see that the region where the LSP gives
the dominant contribution to the matter density of the flat Universe
(marked $\Omega=1$) crosses  the estimated value of
the bottom quark mass ($\mbot=4.25\pm0.1$ GeV) for $\mtop$ broadly
between 160 GeV and 180 GeV. Remarkably, this happens for
150 GeV$\lsim\mhalf,\, \mzero\lsim$ 400 GeV, the range also strongly
favoured
by constraints from $\alpha_s$ and fine tuning.

When the ratio
$\muzero/\mzero$ is decreased, the relic abundance contours
generally move towards larger values of $\mzero$ as do the contours
for $\mtop$ and $\mbot$. For $\muzero/\mzero=1$
the favoured range of the bottom quark mass of about 4.25 GeV lies
entirely within the cosmologically excluded region $\abund>1$.
It also becomes harder to reconcile this region with the fine tuning
constraint and with a value of $\alpha_s(\mz)$ close to 0.122. On
the other hand as $\muzero/\mzero$ increases, $\mbot = 4.25$ takes us
to a region of larger $\mhalf$ and lower $\mzero$ while the contours
relic abundance remain relatively unchanged.
The area consistent with the constraints of $\mbot$, $\mtop$
correspond to lower values of the relic abundance, $\abund \lsim
0.25$.
If we increase $\muzero/
\mzero$ still further the region of overlap for the constraints of
$\mbot$, $\mtop$, and $\Omega=1$ shrinks and leads us the region of
larger fine tuning and smaller $\alpha_s$.
We thus conclude that the combination of all the above constraints
selects the range $1\lsim\muzero/\mzero\lsim {\rm a~few}$.

In the selected range all the Higgs bosons, squarks and sleptons, as
well as the
gluino, are significantly lighter than 1 TeV and thus are bound to
be found at the LHC and SSC.

However, the expectation that the LSP dominates the dark matter
relic density is a natural one. (In minimal supersymmetry no other
particle can even
significantly contribute to the missing mass.) It then implies
a significant {\em lower} bound on the spectrum of supersymmetric
particle masses. We see from Figs.~\ref{figone}d and ~\ref{figtwo}
that the LSP masses favoured by all the constraints lie in the range
\be
60 ~{\rm GeV}\lsim\mchi\lsim 200 ~{\rm GeV},
\label{lsprange}
\ee the upper limit being also expected
in the
minimal supersymmetric model~\cite{chiasdm,bg} on the basis
of naturalness. Similarly, we find
\bea
150 ~{\rm GeV}\lsim\mcharone\lsim300~{\rm GeV}\\
200 ~{\rm GeV}\lsim\msl\lsim500 ~{\rm GeV}\\
250 ~{\rm GeV}\lsim\msq\lsim850 ~{\rm GeV}\\
350 ~{\rm GeV}\lsim\mgluino\lsim900 ~{\rm GeV}.
\label{massranges}
\eea
The heavy Higgs bosons are roughly in the mass range between 250 GeV
and 700 GeV. Of course, lower values of all these masses correspond
to less fine tuning and larger values of $\alpha_s$.
The lightest Higgs boson tree-level mass invariably
comes out close to $\mz$; its one-loop-corrected
value~\cite{radcorrs} is then roughly in the range 120 to 150 GeV.
By comparing Figs.~\ref{figone}b and ~\ref{figtwo}
we also find  $0.116\lsim\alpha_s(\mz)\lsim 0.120$. (Larger values
of $\alpha_s$ are also disfavoured by considering threshold
corrections
at the GUT scale~\cite{meshkov}.)

Thus, if the LSP is indeed the dominant component of DM in the flat
Universe, supersymmetric particles are probably beyond the reach
not only of LEP but also the Tevatron and LEP
200~\cite{chiatlep2,sugradm,dn,texas}.
We note, on the other hand, that smaller ranges of supersymmetric
particles are not firmly excluded but would correspond to the LSP
contributing only a fraction of the critical density. We also note
that we do not claim to have done a fully exhaustive search of the
whole
parameter space. In fact, Drees and Nojiri~\cite{dn} have found in
certain
extreme cases (rather large values of $A_0$) squarks even somewhat
lighter that 200 GeV and a lower limit $\mzero>40$ GeV. We find that
the condition $\Omega=1$ requires in our case significantly larger
values of
$\mzero$ ($\mzero\gsim$ 150 GeV), in agreement with
Refs.~\cite{sugradm,texas}. However, we do not consider it to be in
contradiction with the mentioned results of Ref.~\cite{dn} but a
reflection of somewhat different assumptions at the GUT scale and
methods of deriving the
supersymmetric mass spectra.
We do not expect that the procedure adopted here would produce
substantially modified results by performing a finer search of the
parameter space.


\section{Conclusions}

Our basic conclusions for the
neutralino relic abundance and the associated implications for
the supersymmetric mass spectra are generally consistent with several
other recent analyses.
We do find that cosmologically attractive LSP is almost purely
bino-like ($\sim95\%$) and lies in the range
60 GeV$\lsim\mchi\lsim$ 200 GeV. Moreover, as first noted in
Ref.~\cite{chiatlep2} and confirmed in
Refs.~\cite{sugradm,dn,texas},
if the LSP dominates the dark matter in
the (flat) Universe then the expected ranges of chargino, slepton
and Higgs boson masses lie
beyond the reach of LEP 200.
The associated ranges of gluino and squark
masses
then exceed the reach of the Tevatron but should be discovered at
the SSC and/or the LHC.

Generally, we find it very reassuring that, in the simplest and most
economic supersymmetric scenario, a careful analysis of the
implications
of several
different (and independent) constraints, including
the DM constraint, which result from
the grand unification conditions, leads to
a supersymmetric spectrum accessible to the next generation of
accelerators.

\section*{Acknowledgments}

We thank Graham Ross and Gordon Kane for inspiration and numerous
discussions. This work was supported in part by the US Department
of Energy.

%	\section*{References}
\newpage
%%%%%%%%%%%%% begin refs %%%%%%%%%%%%%%%
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%%%%%%%%%%%%% end of refs %%%%%%%%%%%%%%%
\newpage
\section*{Figure Captions}
\vskip 0.1cm
%	\listoffigures

\begin{figure}[htb]
\vspace{0.1cm}
\caption{In the plane ($\mhalf,\mzero$) for the fixed ratio
$\muzero/\mzero=2$ we show: in window a) the mass contours of the top
and the bottom quarks (solid and short-dashed lines, respectively);
in window b) the contours of $\alpha_s(\mz)$ (solid) and the measure
$c$
of fine-tuning (dots), as discussed in the text; in window c) the
relic
abundance $\abund$ of the LSP; and in window d) the mass contours of
the LSP (solid) and the lightest chargino (dashed) at 50, 100, 150,
200, 500, and 1000 GeV, starting from left,
and the contribution (dots)
of the
bino to the LSP composition (bino purity, as discussed in the text).
In all the windows thick
solid lines delineate regions experimentally excluded by the CDF
(marked CDF) where $\mtop<91$ GeV and by the LEP experiments (LEP)
where the lightest chargino is lighter than 46 GeV. In window c) we
also mark by $\abund>1$ the region cosmologically excluded (too young
Universe). The thin band between the thick dashed lines in window c)
corresponds to the flat Universe ($\Omega=1$), as discussed in the
text. In window d) the region excluded by CDF almost coincides with
the bino purity of 50$\%$ or less.
}
\label{figone}
\end{figure}

\begin{figure}[htb]
\vspace{0.1cm}
\caption{We show a blow-up of the down-left portion of the plane
($\mhalf,\mzero$) from the previous figure for the same fixed ratio
$\muzero/\mzero=2$.  We combine the mass contours of the top and the
bottom quarks with the ones  of the LSP relic mass density. We use
the
same textures as in Fig.~1 but we also show
(two medium-thick short-dashed lines) the contours $\mbot=4.15$ GeV
and
4.35 GeV which reflect the currently favoured range of the mass of
the
bottom quark (see text). We see that they cross the cosmologically
favored region (thick long-dashed lines) marked $\Omega=1$ at roughly
150 GeV$\lsim\mhalf,\, \mzero\lsim$ 400 GeV and for $\mtop$ broadly
between 150 GeV and 180 GeV.
}
\label{figtwo}
\end{figure}
\end{document}
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638 d
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618 d
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632 r
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1098 d
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1150 d
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1220 d
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1234 d
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1226 d
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458 d
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414 d
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464 d
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438 d
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816 d
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1083 d
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900 d
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1156 d
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1217 d
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1415 d
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1484 d
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846 d
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1514 d
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902 1217 d
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1618 d
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d
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1217 d
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896 r
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946 r
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1110 d
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1236 d
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1279 d
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1595 d
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1575 d
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1575 r
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d
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373 2069 r
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2087 d
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2081 d
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2089 d
  399 2091 d  403 2091 d  409 2089 d  413 2085 d  415 2081 d  373
2061 r
  373 2069 d  365 2105 r  369 2109 d  375 2113 d  383 2117 d  393
2119 d
  401 2119 d  411 2117 d  419 2113 d  425 2109 d  429 2105 d  369
2109 r
  377 2113 d  383 2115 d  393 2117 d  401 2117 d  411 2115 d  417
2113 d
  425 2109 d   2 lw  979 2002 r  979 2032 d  978 2002 r  978 2032 d
983 2002 r
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2002 r
  952 2032 d  951 2002 r  951 2032 d  943 2011 r  943 2022 d  957
2002 r
  934 2002 d  934 2009 d  936 2002 d  951 2016 r  943 2016 d  957
2032 r
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2002 r
  922 2032 d  927 2002 r  910 2002 d  906 2004 d  905 2005 d  903
2008 d
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2002 r
  908 2004 d  906 2005 d  905 2008 d  905 2012 d  906 2015 d  908
2016 d
  910 2018 d  927 2032 r  917 2032 d  871 2718 r  876 2721 d  865
2721 d
  871 2718 d  867 2714 d  865 2707 d  865 2703 d  867 2696 d  871
2692 d
  876 2690 d  882 2688 d  893 2688 d  900 2690 d  904 2692 d  909
2696 d
  911 2703 d  911 2707 d  909 2714 d  904 2718 d  900 2721 d  865
2703 r
  867 2699 d  871 2694 d  876 2692 d  882 2690 d  893 2690 d  900
2692 d
  904 2694 d  909 2699 d  911 2703 d  865 2738 r  911 2738 d  865
2740 r
  911 2740 d  865 2732 r  865 2754 d  867 2760 d  871 2765 d  876
2767 d
  882 2769 d  893 2769 d  900 2767 d  904 2765 d  909 2760 d  911
2754 d
  911 2732 d  865 2754 r  867 2758 d  871 2762 d  876 2765 d  882
2767 d
  893 2767 d  900 2765 d  904 2762 d  909 2758 d  911 2754 d  865
2787 r
  911 2787 d  865 2789 r  911 2789 d  878 2802 r  895 2802 d  865
2780 r
  865 2815 d  876 2815 d  865 2813 d  887 2789 r  887 2802 d  911
2780 r
  911 2795 d   1 lw  941 2196 r  939 2199 d  935 2203 d  965 2203 d
937 2202 r
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2222 d
  948 2220 d  952 2220 d  959 2222 d  963 2224 d  965 2229 d  965
2231 d
  963 2236 d  959 2238 d  952 2240 d  948 2240 d  941 2238 d  937
2236 d
  935 2231 d  935 2229 d  937 2226 d  938 2224 d  941 2223 d  948
2222 d
  952 2222 d  959 2223 d  962 2224 d  963 2226 d  965 2229 d  965
2231 r
  963 2234 d  962 2236 d  959 2237 d  952 2238 d  948 2238 d  941
2237 d
  938 2236 d  937 2234 d  935 2231 d  842 2194 r  843 2195 d  845
2194 d
  843 2192 d  842 2192 d  839 2194 d  838 2195 d  836 2199 d  836
2205 d
  838 2209 d  839 2210 d  842 2212 d  845 2212 d  847 2210 d  850
2206 d
  853 2199 d  854 2196 d  857 2194 d  861 2192 d  866 2192 d  836
2205 r
  838 2208 d  839 2209 d  842 2210 d  845 2210 d  847 2209 d  850
2205 d
  853 2199 d  863 2192 r  861 2194 d  861 2196 d  864 2203 d  864
2208 d
  863 2210 d  861 2212 d  861 2196 r  866 2203 d  866 2209 d  864
2210 d
  861 2212 d  859 2212 d  836 2229 r  838 2224 d  842 2222 d  849
2220 d
  853 2220 d  860 2222 d  864 2224 d  866 2229 d  866 2231 d  864
2236 d
  860 2238 d  853 2240 d  849 2240 d  842 2238 d  838 2236 d  836
2231 d
  836 2229 d  838 2226 d  839 2224 d  842 2223 d  849 2222 d  853
2222 d
  860 2223 d  863 2224 d  864 2226 d  866 2229 d  866 2231 r  864
2234 d
  863 2236 d  860 2237 d  853 2238 d  849 2238 d  842 2237 d  839
2236 d
  838 2234 d  836 2231 d  722 2195 r  736 2192 d  735 2195 d  733
2199 d
  733 2203 d  735 2208 d  738 2210 d  742 2212 d  743 2212 d  747
2210 d
  750 2208 d  752 2203 d  752 2199 d  750 2195 d  749 2194 d  746
2192 d
  745 2192 d  743 2194 d  745 2195 d  746 2194 d  733 2203 r  735
2206 d
  738 2209 d  742 2210 d  743 2210 d  747 2209 d  750 2206 d  752
2203 d
  722 2195 r  722 2209 d  724 2195 r  724 2202 d  722 2209 d  722
2229 r
  724 2224 d  728 2222 d  735 2220 d  739 2220 d  746 2222 d  750
2224 d
  752 2229 d  752 2231 d  750 2236 d  746 2238 d  739 2240 d  735
2240 d
  728 2238 d  724 2236 d  722 2231 d  722 2229 d  724 2226 d  725
2224 d
  728 2223 d  735 2222 d  739 2222 d  746 2223 d  749 2224 d  750
2226 d
  752 2229 d  752 2231 r  750 2234 d  749 2236 d  746 2237 d  739
2238 d
  735 2238 d  728 2237 d  725 2236 d  724 2234 d  722 2231 d  660
2182 r
  658 2185 d  654 2189 d  684 2189 d  656 2188 r  684 2188 d  684
2182 r
  684 2195 d  654 2215 r  656 2210 d  660 2208 d  667 2206 d  671
2206 d
  678 2208 d  682 2210 d  684 2215 d  684 2217 d  682 2222 d  678
2224 d
  671 2226 d  667 2226 d  660 2224 d  656 2222 d  654 2217 d  654
2215 d
  656 2212 d  657 2210 d  660 2209 d  667 2208 d  671 2208 d  678
2209 d
  681 2210 d  682 2212 d  684 2215 d  684 2217 r  682 2220 d  681
2222 d
  678 2223 d  671 2224 d  667 2224 d  660 2223 d  657 2222 d  656
2220 d
  654 2217 d  654 2243 r  656 2238 d  660 2236 d  667 2234 d  671
2234 d
  678 2236 d  682 2238 d  684 2243 d  684 2245 d  682 2250 d  678
2252 d
  671 2254 d  667 2254 d  660 2252 d  656 2250 d  654 2245 d  654
2243 d
  656 2240 d  657 2238 d  660 2237 d  667 2236 d  671 2236 d  678
2237 d
  681 2238 d  682 2240 d  684 2243 d  684 2245 r  682 2248 d  681
2250 d
  678 2251 d  671 2252 d  667 2252 d  660 2251 d  657 2250 d  656
2248 d
  654 2245 d  584 2180 r  585 2181 d  587 2180 d  585 2178 d  584
2178 d
  581 2180 d  580 2181 d  578 2185 d  578 2191 d  580 2195 d  581
2196 d
  584 2198 d  587 2198 d  589 2196 d  592 2192 d  595 2185 d  596
2182 d
  599 2180 d  603 2178 d  608 2178 d  578 2191 r  580 2194 d  581
2195 d
  584 2196 d  587 2196 d  589 2195 d  592 2191 d  595 2185 d  605
2178 r
  603 2180 d  603 2182 d  606 2189 d  606 2194 d  605 2196 d  603
2198 d
  603 2182 r  608 2189 d  608 2195 d  606 2196 d  603 2198 d  601
2198 d
  578 2215 r  580 2210 d  584 2208 d  591 2206 d  595 2206 d  602
2208 d
  606 2210 d  608 2215 d  608 2217 d  606 2222 d  602 2224 d  595
2226 d
  591 2226 d  584 2224 d  580 2222 d  578 2217 d  578 2215 d  580
2212 d
  581 2210 d  584 2209 d  591 2208 d  595 2208 d  602 2209 d  605
2210 d
  606 2212 d  608 2215 d  608 2217 r  606 2220 d  605 2222 d  602
2223 d
  595 2224 d  591 2224 d  584 2223 d  581 2222 d  580 2220 d  578
2217 d
  578 2243 r  580 2238 d  584 2236 d  591 2234 d  595 2234 d  602
2236 d
  606 2238 d  608 2243 d  608 2245 d  606 2250 d  602 2252 d  595
2254 d
  591 2254 d  584 2252 d  580 2250 d  578 2245 d  578 2243 d  580
2240 d
  581 2238 d  584 2237 d  591 2236 d  595 2236 d  602 2237 d  605
2238 d
  606 2240 d  608 2243 d  608 2245 r  606 2248 d  605 2250 d  602
2251 d
  595 2252 d  591 2252 d  584 2251 d  581 2250 d  580 2248 d  578
2245 d
  494 2181 r  508 2178 d  507 2181 d  505 2185 d  505 2189 d  507
2194 d
  510 2196 d  514 2198 d  515 2198 d  519 2196 d  522 2194 d  524
2189 d
  524 2185 d  522 2181 d  521 2180 d  518 2178 d  517 2178 d  515
2180 d
  517 2181 d  518 2180 d  505 2189 r  507 2192 d  510 2195 d  514
2196 d
  515 2196 d  519 2195 d  522 2192 d  524 2189 d  494 2181 r  494
2195 d
  496 2181 r  496 2188 d  494 2195 d  494 2215 r  496 2210 d  500
2208 d
  507 2206 d  511 2206 d  518 2208 d  522 2210 d  524 2215 d  524
2217 d
  522 2222 d  518 2224 d  511 2226 d  507 2226 d  500 2224 d  496
2222 d
  494 2217 d  494 2215 d  496 2212 d  497 2210 d  500 2209 d  507
2208 d
  511 2208 d  518 2209 d  521 2210 d  522 2212 d  524 2215 d  524
2217 r
  522 2220 d  521 2222 d  518 2223 d  511 2224 d  507 2224 d  500
2223 d
  497 2222 d  496 2220 d  494 2217 d  494 2243 r  496 2238 d  500
2236 d
  507 2234 d  511 2234 d  518 2236 d  522 2238 d  524 2243 d  524
2245 d
  522 2250 d  518 2252 d  511 2254 d  507 2254 d  500 2252 d  496
2250 d
  494 2245 d  494 2243 d  496 2240 d  497 2238 d  500 2237 d  507
2236 d
  511 2236 d  518 2237 d  521 2238 d  522 2240 d  524 2243 d  524
2245 r
  522 2248 d  521 2250 d  518 2251 d  511 2252 d  507 2252 d  500
2251 d
  497 2250 d  496 2248 d  494 2245 d  439 2168 r  437 2171 d  433
2175 d
  463 2175 d  435 2174 r  463 2174 d  463 2168 r  463 2181 d  433
2201 r
  435 2196 d  439 2194 d  446 2192 d  450 2192 d  457 2194 d  461
2196 d
  463 2201 d  463 2203 d  461 2208 d  457 2210 d  450 2212 d  446
2212 d
  439 2210 d  435 2208 d  433 2203 d  433 2201 d  435 2198 d  436
2196 d
  439 2195 d  446 2194 d  450 2194 d  457 2195 d  460 2196 d  461
2198 d
  463 2201 d  463 2203 r  461 2206 d  460 2208 d  457 2209 d  450
2210 d
  446 2210 d  439 2209 d  436 2208 d  435 2206 d  433 2203 d  433
2229 r
  435 2224 d  439 2222 d  446 2220 d  450 2220 d  457 2222 d  461
2224 d
  463 2229 d  463 2231 d  461 2236 d  457 2238 d  450 2240 d  446
2240 d
  439 2238 d  435 2236 d  433 2231 d  433 2229 d  435 2226 d  436
2224 d
  439 2223 d  446 2222 d  450 2222 d  457 2223 d  460 2224 d  461
2226 d
  463 2229 d  463 2231 r  461 2234 d  460 2236 d  457 2237 d  450
2238 d
  446 2238 d  439 2237 d  436 2236 d  435 2234 d  433 2231 d  433
2257 r
  435 2252 d  439 2250 d  446 2248 d  450 2248 d  457 2250 d  461
2252 d
  463 2257 d  463 2259 d  461 2264 d  457 2266 d  450 2268 d  446
2268 d
  439 2266 d  435 2264 d  433 2259 d  433 2257 d  435 2254 d  436
2252 d
  439 2251 d  446 2250 d  450 2250 d  457 2251 d  460 2252 d  461
2254 d
  463 2257 d  463 2259 r  461 2262 d  460 2264 d  457 2265 d  450
2266 d
  446 2266 d  439 2265 d  436 2264 d  435 2262 d  433 2259 d  884
2259 r
  885 2255 d  888 2252 d  891 2251 d  895 2249 d  899 2249 d  902
2251 d
  904 2255 d  904 2258 d  902 2260 d  898 2265 d  894 2267 d  888
2270 d
  884 2272 d  884 2259 r  885 2256 d  888 2253 d  891 2252 d  895
2251 d
  899 2251 d  902 2252 d  904 2255 d  884 2259 r  884 2262 d  885
2266 d
  890 2267 d  898 2269 d  902 2270 d  904 2272 d  884 2262 r  885
2265 d
  890 2266 d  898 2267 d  902 2269 d  904 2272 d  904 2273 d  901
2288 r
  900 2289 d  903 2289 d  901 2288 d  900 2286 d  900 2283 d  901
2281 d
  902 2280 d  903 2280 d  904 2281 d  905 2282 d  907 2286 d  908
2288 d
  909 2289 d  903 2280 r  904 2281 d  904 2282 d  906 2286 d  907
2288 d
  909 2289 d  910 2289 d  911 2288 d  912 2286 d  912 2283 d  911
2281 d
  909 2280 d  912 2280 d  911 2281 d  887 2296 r  887 2321 d  895
2296 r
  895 2321 d  874 2338 r  876 2333 d  880 2331 d  887 2329 d  891
2329 d
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2724 d
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2740 d
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2780 d
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2800 d
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2794 r
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2802 d
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2820 d
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2834 d
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2840 d
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2820 d
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2834 r
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2842 d
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1982 r
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2073 r
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2116 r
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2383 r
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2491 r
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2560 r
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2772 r
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2906 r
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1982 r
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1982 r
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1982 r
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1982 r
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1982 r
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1982 r
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1850 r
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1878 d
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1892 d
 1021 1898 d 1015 1902 d 1005 1904 d  999 1904 d  989 1902 d  983
1898 d
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1878 d
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1892 r
 1021 1896 d 1019 1898 d 1015 1900 d 1005 1902 d  999 1902 d  989
1900 d
  985 1898 d  983 1896 d  981 1892 d  981 1928 r  983 1922 d  989
1918 d
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1932 d
 1021 1938 d 1015 1942 d 1005 1944 d  999 1944 d  989 1942 d  983
1938 d
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1918 d
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1932 r
 1021 1936 d 1019 1938 d 1015 1940 d 1005 1942 d  999 1942 d  989
1940 d
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1838 d
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1854 d
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1856 d
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1854 r
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1854 d
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1858 d
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1862 d
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1876 d
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1898 d
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1892 d
  854 1888 d  856 1884 d  858 1882 d  862 1880 d  872 1878 d  878
1878 d
  888 1880 d  892 1882 d  894 1884 d  896 1888 d  896 1892 r  894
1896 d
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1898 d
  856 1896 d  854 1892 d  854 1928 r  856 1922 d  862 1918 d  872
1916 d
  878 1916 d  888 1918 d  894 1922 d  896 1928 d  896 1932 d  894
1938 d
  888 1942 d  878 1944 d  872 1944 d  862 1942 d  856 1938 d  854
1932 d
  854 1928 d  856 1924 d  858 1922 d  862 1920 d  872 1918 d  878
1918 d
  888 1920 d  892 1922 d  894 1924 d  896 1928 d  896 1932 r  894
1936 d
  892 1938 d  888 1940 d  878 1942 d  872 1942 d  862 1940 d  858
1938 d
  856 1936 d  854 1932 d  687 1840 r  707 1836 d  705 1840 d  703
1846 d
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1862 d
  727 1858 d  729 1852 d  729 1846 d  727 1840 d  725 1838 d  721
1836 d
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1856 d
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1852 d
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1888 r
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1882 d
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1904 d
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1882 d
  695 1880 d  705 1878 d  711 1878 d  721 1880 d  725 1882 d  727
1884 d
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1902 d
  705 1902 d  695 1900 d  691 1898 d  689 1896 d  687 1892 d  687
1928 r
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1922 d
  729 1928 d  729 1932 d  727 1938 d  721 1942 d  711 1944 d  705
1944 d
  695 1942 d  689 1938 d  687 1932 d  687 1928 d  689 1924 d  691
1922 d
  695 1920 d  705 1918 d  711 1918 d  721 1920 d  725 1922 d  727
1924 d
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1942 d
  705 1942 d  695 1940 d  691 1938 d  689 1936 d  687 1932 d  568
1802 r
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1802 r
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1836 d
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1862 d
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1848 d
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1840 d
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1858 d
  594 1860 d  584 1862 d  578 1862 d  568 1860 d  564 1858 d  562
1856 d
  560 1852 d  560 1888 r  562 1882 d  568 1878 d  578 1876 d  584
1876 d
  594 1878 d  600 1882 d  602 1888 d  602 1892 d  600 1898 d  594
1902 d
  584 1904 d  578 1904 d  568 1902 d  562 1898 d  560 1892 d  560
1888 d
  562 1884 d  564 1882 d  568 1880 d  578 1878 d  584 1878 d  594
1880 d
  598 1882 d  600 1884 d  602 1888 d  602 1892 r  600 1896 d  598
1898 d
  594 1900 d  584 1902 d  578 1902 d  568 1900 d  564 1898 d  562
1896 d
  560 1892 d  560 1928 r  562 1922 d  568 1918 d  578 1916 d  584
1916 d
  594 1918 d  600 1922 d  602 1928 d  602 1932 d  600 1938 d  594
1942 d
  584 1944 d  578 1944 d  568 1942 d  562 1938 d  560 1932 d  560
1928 d
  562 1924 d  564 1922 d  568 1920 d  578 1918 d  584 1918 d  594
1920 d
  598 1922 d  600 1924 d  602 1928 d  602 1932 r  600 1936 d  598
1938 d
  594 1940 d  584 1942 d  578 1942 d  568 1940 d  564 1938 d  562
1936 d
  560 1932 d  442 1798 r  444 1800 d  446 1798 d  444 1796 d  442
1796 d
  438 1798 d  436 1800 d  434 1806 d  434 1814 d  436 1820 d  438
1822 d
  442 1824 d  446 1824 d  450 1822 d  454 1816 d  458 1806 d  460
1802 d
  464 1798 d  470 1796 d  476 1796 d  434 1814 r  436 1818 d  438
1820 d
  442 1822 d  446 1822 d  450 1820 d  454 1814 d  458 1806 d  472
1796 r
  470 1798 d  470 1802 d  474 1812 d  474 1818 d  472 1822 d  470
1824 d
  470 1802 r  476 1812 d  476 1820 d  474 1822 d  470 1824 d  466
1824 d
  434 1848 r  436 1842 d  442 1838 d  452 1836 d  458 1836 d  468
1838 d
  474 1842 d  476 1848 d  476 1852 d  474 1858 d  468 1862 d  458
1864 d
  452 1864 d  442 1862 d  436 1858 d  434 1852 d  434 1848 d  436
1844 d
  438 1842 d  442 1840 d  452 1838 d  458 1838 d  468 1840 d  472
1842 d
  474 1844 d  476 1848 d  476 1852 r  474 1856 d  472 1858 d  468
1860 d
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1852 d
  434 1888 r  436 1882 d  442 1878 d  452 1876 d  458 1876 d  468
1878 d
  474 1882 d  476 1888 d  476 1892 d  474 1898 d  468 1902 d  458
1904 d
  452 1904 d  442 1902 d  436 1898 d  434 1892 d  434 1888 d  436
1884 d
  438 1882 d  442 1880 d  452 1878 d  458 1878 d  468 1880 d  472
1882 d
  474 1884 d  476 1888 d  476 1892 r  474 1896 d  472 1898 d  468
1900 d
  458 1902 d  452 1902 d  442 1900 d  438 1898 d  436 1896 d  434
1892 d
  434 1928 r  436 1922 d  442 1918 d  452 1916 d  458 1916 d  468
1918 d
  474 1922 d  476 1928 d  476 1932 d  474 1938 d  468 1942 d  458
1944 d
  452 1944 d  442 1942 d  436 1938 d  434 1932 d  434 1928 d  436
1924 d
  438 1922 d  442 1920 d  452 1918 d  458 1918 d  468 1920 d  472
1922 d
  474 1924 d  476 1928 d  476 1932 r  474 1936 d  472 1938 d  468
1940 d
  458 1942 d  452 1942 d  442 1940 d  438 1938 d  436 1936 d  434
1932 d
  266 1800 r  286 1796 d  284 1800 d  282 1806 d  282 1812 d  284
1818 d
  288 1822 d  294 1824 d  296 1824 d  302 1822 d  306 1818 d  308
1812 d
  308 1806 d  306 1800 d  304 1798 d  300 1796 d  298 1796 d  296
1798 d
  298 1800 d  300 1798 d  282 1812 r  284 1816 d  288 1820 d  294
1822 d
  296 1822 d  302 1820 d  306 1816 d  308 1812 d  266 1800 r  266
1820 d
  268 1800 r  268 1810 d  266 1820 d  266 1848 r  268 1842 d  274
1838 d
  284 1836 d  290 1836 d  300 1838 d  306 1842 d  308 1848 d  308
1852 d
  306 1858 d  300 1862 d  290 1864 d  284 1864 d  274 1862 d  268
1858 d
  266 1852 d  266 1848 d  268 1844 d  270 1842 d  274 1840 d  284
1838 d
  290 1838 d  300 1840 d  304 1842 d  306 1844 d  308 1848 d  308
1852 r
  306 1856 d  304 1858 d  300 1860 d  290 1862 d  284 1862 d  274
1860 d
  270 1858 d  268 1856 d  266 1852 d  266 1888 r  268 1882 d  274
1878 d
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1892 d
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2054 d
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2918 d
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r
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2638 r
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2856 r
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2824 d
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2869 r
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2494 r
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2249 d
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2383 d
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lt0
 1005 2087 r  878 2044 d  752 2010 d  625 1997 d  498 1992 d  372
1990 d
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2127 1654 r
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805 d
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725 d
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1075 d
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1086 d
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 1740 1114 d 1710 1127 r 1712 1130 d 1713 1131 d 1716 1133 d 1719
1133 d
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1955  682 r
 1923  682 d 1955  717 r 1923  717 d 1955  747 r 1923  747 d 1955
773 r
 1923  773 d 1955  795 r 1923  795 d 1955  816 r 1923  816 d 1955
816 r
 1891  816 d 1955  949 r 1923  949 d 1955 1028 r 1891 1028 d 1955
1083 r
 1923 1083 d 1955 1126 r 1923 1126 d 1955 1161 r 1923 1161 d 1955
1191 r
 1923 1191 d 1955 1217 r 1923 1217 d 1955 1240 r 1923 1240 d 1955
1260 r
 1923 1260 d 1955 1260 r 1891 1260 d 1955 1394 r 1923 1394 d 1955
1472 r
 1891 1472 d 1955 1527 r 1923 1527 d 1955 1570 r 1923 1570 d 1955
1606 r
 1923 1606 d 2003  768 r 2001  772 d 1995  778 d 2037  778 d 1997
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 2037  776 d 2037  768 r 2037  786 d 1995  814 r 1997  808 d 2003
804 d
 2013  802 d 2019  802 d 2029  804 d 2035  808 d 2037  814 d 2037
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 2035  824 d 2029  828 d 2019  830 d 2013  830 d 2003  828 d 1997
824 d
 1995  818 d 1995  814 d 1997  810 d 1999  808 d 2003  806 d 2013
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 2019  804 d 2029  806 d 2033  808 d 2035  810 d 2037  814 d 2037
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 2035  822 d 2033  824 d 2029  826 d 2019  828 d 2013  828 d 2003
826 d
 1999  824 d 1997  822 d 1995  818 d 1995  854 r 1997  848 d 2003
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 2013  842 d 2019  842 d 2029  844 d 2035  848 d 2037  854 d 2037
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 2035  864 d 2029  868 d 2019  870 d 2013  870 d 2003  868 d 1997
864 d
 1995  858 d 1995  854 d 1997  850 d 1999  848 d 2003  846 d 2013
844 d
 2019  844 d 2029  846 d 2033  848 d 2035  850 d 2037  854 d 2037
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 2035  862 d 2033  864 d 2029  866 d 2019  868 d 2013  868 d 2003
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 1999  864 d 1997  862 d 1995  858 d 2001  976 r 2003  978 d 2005
976 d
 2003  974 d 2001  974 d 1997  978 d 1995  984 d 1995  992 d 1997
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 2001 1000 d 2007 1000 d 2011  998 d 2013  992 d 2013  986 d 1995
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 2019 1000 d 2023 1002 d 2029 1002 d 2033 1000 d 2035  998 d 2037
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1030 d
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1070 d
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1232 d
 2003 1228 d 2013 1226 d 2019 1226 d 2029 1228 d 2035 1232 d 2037
1238 d
 2037 1242 d 2035 1248 d 2029 1252 d 2019 1254 d 2013 1254 d 2003
1252 d
 1997 1248 d 1995 1242 d 1995 1238 d 1997 1234 d 1999 1232 d 2003
1230 d
 2013 1228 d 2019 1228 d 2029 1230 d 2033 1232 d 2035 1234 d 2037
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 2037 1242 r 2035 1246 d 2033 1248 d 2029 1250 d 2019 1252 d 2013
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 2003 1268 d 2013 1266 d 2019 1266 d 2029 1268 d 2035 1272 d 2037
1278 d
 2037 1282 d 2035 1288 d 2029 1292 d 2019 1294 d 2013 1294 d 2003
1292 d
 1997 1288 d 1995 1282 d 1995 1278 d 1997 1274 d 1999 1272 d 2003
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 2013 1268 d 2019 1268 d 2029 1270 d 2033 1272 d 2035 1274 d 2037
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 2037 1282 r 2035 1286 d 2033 1288 d 2029 1290 d 2019 1292 d 2013
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 2003 1290 d 1999 1288 d 1997 1286 d 1995 1282 d 1995 1318 r 1997
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 2003 1308 d 2013 1306 d 2019 1306 d 2029 1308 d 2035 1312 d 2037
1318 d
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1332 d
 1997 1328 d 1995 1322 d 1995 1318 d 1997 1314 d 1999 1312 d 2003
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 2013 1308 d 2019 1308 d 2029 1310 d 2033 1312 d 2035 1314 d 2037
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 2037 1322 r 2035 1326 d 2033 1328 d 2029 1330 d 2019 1332 d 2013
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 2003 1330 d 1999 1328 d 1997 1326 d 1995 1322 d 2001 1400 r 2003
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1410 d
 1995 1416 r 1997 1420 d 2001 1422 d 2007 1422 d 2011 1420 d 2013
1416 d
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 2025 1400 d 2027 1402 d 2029 1400 d 2017 1422 r 2023 1424 d 2029
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 2013 1438 d 2019 1438 d 2029 1440 d 2035 1444 d 2037 1450 d 2037
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 2035 1460 d 2029 1464 d 2019 1466 d 2013 1466 d 2003 1464 d 1997
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 1995 1454 d 1995 1450 d 1997 1446 d 1999 1444 d 2003 1442 d 2013
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 2019 1440 d 2029 1442 d 2033 1444 d 2035 1446 d 2037 1450 d 2037
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 1999 1460 d 1997 1458 d 1995 1454 d 1995 1490 r 1997 1484 d 2003
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 2013 1478 d 2019 1478 d 2029 1480 d 2035 1484 d 2037 1490 d 2037
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 2035 1500 d 2029 1504 d 2019 1506 d 2013 1506 d 2003 1504 d 1997
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 1995 1494 d 1995 1490 d 1997 1486 d 1999 1484 d 2003 1482 d 2013
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 2019 1480 d 2029 1482 d 2033 1484 d 2035 1486 d 2037 1490 d 2037
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 2013 1518 d 2019 1518 d 2029 1520 d 2035 1524 d 2037 1530 d 2037
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 1995 1534 d 1995 1530 d 1997 1526 d 1999 1524 d 2003 1522 d 2013
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 2035 1538 d 2033 1540 d 2029 1542 d 2019 1544 d 2013 1544 d 2003
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550 r
 1973  550 d 1973  542 r 1973  560 d 1931  588 r 1933  582 d 1939
578 d
 1949  576 d 1955  576 d 1965  578 d 1971  582 d 1973  588 d 1973
592 d
 1971  598 d 1965  602 d 1955  604 d 1949  604 d 1939  602 d 1933
598 d
 1931  592 d 1931  588 d 1933  584 d 1935  582 d 1939  580 d 1949
578 d
 1955  578 d 1965  580 d 1969  582 d 1971  584 d 1973  588 d 1973
592 r
 1971  596 d 1969  598 d 1965  600 d 1955  602 d 1949  602 d 1939
600 d
 1935  598 d 1933  596 d 1931  592 d 1931  628 r 1933  622 d 1939
618 d
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632 d
 1971  638 d 1965  642 d 1955  644 d 1949  644 d 1939  642 d 1933
638 d
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618 d
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632 r
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 1806  558 d 1808  560 d 1812  562 d 1816  562 d 1820  560 d 1824
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 1828  546 d 1842  536 r 1840  538 d 1840  542 d 1844  552 d 1844
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 1828  576 d 1838  578 d 1844  582 d 1846  588 d 1846  592 d 1844
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1900 d
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1918 d
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1800 r
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652 r
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652 r
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1038 d
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 2003 1052 d 2013 1050 d 2019 1050 d 2029 1052 d 2035 1056 d 2037
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1962 d
 2087 1962 d 2097 1960 d 2103 1958 d 2111 1954 d 1244  432 r 1244
460 d
 1242  432 r 1242  460 d 1242  438 r 1238  434 d 1232  432 d 1228
432 d
 1222  434 d 1220  438 d 1220  460 d 1228  432 r 1224  434 d 1222
438 d
 1222  460 d 1220  438 r 1216  434 d 1210  432 d 1206  432 d 1200
434 d
 1198  438 d 1198  460 d 1206  432 r 1202  434 d 1200  438 d 1200
460 d
 1250  432 r 1242  432 d 1250  460 r 1236  460 d 1228  460 r 1214
460 d
 1206  460 r 1192  460 d 1177  447 r 1181  448 d 1183  452 d 1184
458 d
 1184  461 d 1183  467 d 1181  471 d 1177  472 d 1175  472 d 1171
471 d
 1169  467 d 1168  461 d 1168  458 d 1169  452 d 1171  448 d 1175
447 d
 1177  447 d 1180  448 d 1181  449 d 1182  452 d 1183  458 d 1183
461 d
 1182  467 d 1181  470 d 1180  471 d 1177  472 d 1175  472 r 1172
471 d
 1171  470 d 1170  467 d 1169  461 d 1169  458 d 1170  452 d 1171
449 d
 1172  448 d 1175  447 d 1106  410 r 1110  414 d 1114  420 d 1118
428 d
 1120  438 d 1120  446 d 1118  456 d 1114  464 d 1110  470 d 1106
474 d
 1110  414 r 1114  422 d 1116  428 d 1118  438 d 1118  446 d 1116
456 d
 1114  462 d 1110  470 d 1066  424 r 1064  428 d 1064  418 d 1066
424 d
 1070  420 d 1076  418 d 1080  418 d 1086  420 d 1090  424 d 1092
428 d
 1094  434 d 1094  444 d 1092  450 d 1090  454 d 1086  458 d 1080
460 d
 1076  460 d 1070  458 d 1066  454 d 1080  418 r 1084  420 d 1088
424 d
 1090  428 d 1092  434 d 1092  444 d 1090  450 d 1088  454 d 1084
458 d
 1080  460 d 1066  444 r 1066  460 d 1064  444 r 1064  460 d 1072
444 r
 1060  444 d 1048  444 r 1024  444 d 1024  440 d 1026  436 d 1028
434 d
 1032  432 d 1038  432 d 1044  434 d 1048  438 d 1050  444 d 1050
448 d
 1048  454 d 1044  458 d 1038  460 d 1034  460 d 1028  458 d 1024
454 d
 1026  444 r 1026  438 d 1028  434 d 1038  432 r 1042  434 d 1046
438 d
 1048  444 d 1048  448 d 1046  454 d 1042  458 d 1038  460 d 1010
418 r
  996  460 d 1008  418 r  996  454 d  982  418 r  996  460 d 1016
418 r
 1002  418 d  988  418 r  976  418 d  968  410 r  964  414 d  960
420 d
  956  428 d  954  438 d  954  446 d  956  456 d  960  464 d  964
470 d
  968  474 d  964  414 r  960  422 d  958  428 d  956  438 d  956
446 d
  958  456 d  960  462 d  964  470 d 1955 1277 r 1683 1427 d 1527
1516 d
 1955 1964 r 1690 2173 d 1955 2422 r 1809 2545 d 1690 2173 r 1527
2309 d
 1809 2545 r 1527 2798 d 1527 1516 r 1100 1720 d 1527 2309 r 1236
2545 d
 1527 2798 r 1392 2918 d 1236 2545 r 1100 2666 d 1100 1720 r  853
1800 d
  672 1859 d 1100 2666 r  794 2918 d  672 1859 r  245 1921 d lt2 1955
749 r
 1527  973 d 1955 1482 r 1661 1800 d 1527 1953 d 1955 2180 r 1748
2545 d
 1527 2885 d 1527  973 r 1398 1055 d 1100 1272 d 1527 1953 r 1341
2173 d
 1100 2430 d 1527 2885 r 1502 2918 d 1100 1272 r  885 1427 d  672
1675 d
 1100 2430 r  994 2545 d  672 2891 d  672 1675 r  554 1800 d  245
2156 d
  672 2891 r  646 2918 d   5 lw 1955 1056 r 1527 1380 d 1955 1237 r
1712 1427 d
 1527 1623 d 1527 1380 r 1466 1427 d 1101 1800 d 1527 1623 r 1350
1800 d
 1101 1800 r 1100 1801 d 1350 1800 r 1100 2096 d 1100 1801 r  768
2173 d
 1100 2096 r 1031 2173 d  768 2173 r  672 2279 d 1031 2173 r  695
2545 d
  672 2570 d  672 2279 r  420 2545 d  245 2722 d  672 2570 r  325
2918 d
   9 lw lt0 1955 1294 r 1527 1206 d 1100 1092 d  672  964 r  941 1055
d
 1100 1092 r  941 1055 d  672  964 r  245  867 d  283  682 r  252
1055 d
  245 1060 d  245 1693 r  401 1800 d  656 2173 d  245 2455 d   7 lw
lt3
 1955  689 r 1948  682 d 1100 1864 r 1423 2173 d 1100 2497 d  764
682 r
  901 1055 d  672 1257 d  672 1534 r 1035 1800 d  672 1803 r 1069
2173 d
 1100 1864 r 1035 1800 d 1069 2173 r  672 2467 d 1100 2497 r 1014
2545 d
  768 2918 d  558  682 r  585 1055 d  245 1316 d  672 1257 r  354
1427 d
  245 1507 r  670 1800 d  672 1534 r  354 1427 d  672 1803 r  670
1800 d
  672 2467 r  479 2545 d  296 2918 d
e
%%Trailer
EndPSPlot

