%Paper: 
%From: "Lawrence Krauss, 203-432-6923"
% <KRAUSS%YALPH2.bitnet@yalevm.ycc.yale.edu>
%Date: Fri, 8 May 1992 15:48 EDT


\documentstyle[12pt]{article}
\textwidth7in
\oddsidemargin-.2in
\begin{document}
\baselineskip=21pt
\rightline{YCTP-P15-92}
\rightline{April 1992}
\vskip .2in
\begin{center}
{\large{\bf Grand Unification, Gravitational Waves, and
the Cosmic Microwave Background Anisotropy}}
\end{center}
\vskip .1in
\begin{center}
Lawrence M.
Krauss\footnote{Also Dept. of Astronomy.  Research supported in part
by the NSF,DOE, and TNRLC}
and Martin White\footnote{Address after September 1, Center for Particle
Astrophysics, Berkeley}

{\it Center for Theoretical Physics, Sloane
Laboratory}

{\it  Yale University, New Haven, CT 06511}
\end{center}

\vskip .2in

\centerline{ {\bf Abstract} }

\baselineskip=18pt

\noindent
We re-examine the gravitational wave background resulting from inflation and
its effect on the cosmic microwave background radiation. The new COBE
measurement of a cosmic background quadrupole anisotropy places an upper limit
on the vacuum energy during inflation of $\approx 5 \times 10^{16}$ GeV.
A stochastic background of gravitational waves from inflation could produce
the entire observed signal (consistent with the observed dipole anisotropy and
a flat spectrum) if the vacuum energy during inflation was as small as
$1.5 \times 10^{16}$ GeV at the 95\% confidence level.
This coincides nicely with the mass scale for Grand Unification inferred from
precision measurements of the electroweak and strong coupling constants, for
the SUSY Grand Unified Theories.  Thus COBE could be providing the first direct
evidence, via gravitational waves, for GUTs, and supersymmetry. Further tests
of this possibility are examined, based on analyzing the energy density
associated with gravitational waves from inflation.

\baselineskip=21pt

\newpage

The observation by the DMR instrument aboard the COBE satellite of
large scale anisotropies in the Cosmic Microwave
background (CMB) \cite{smoot} is probably the most important discovery in
cosmology since the discovery of the CMB itself \cite{penzias}.  Such
anisotropies cannot have been induced by causal processes which were
initiated after the era of recombination and thus represent true
primordial fluctuations resulting from physics associated with the
initial conditions of the FRW cosmology itself.  These initial
conditions are likely to have resulted from processes associated with
either an inflationary phase or new planck scale physics.  Only in the former
case can explicit predictions be made and the COBE data on the temperature
correlation function is remarkably consistent with a flat Harrison-Zel'dovich
spectrum as predicted from inflation.

Inflation predicts at least two sources of Harrison-Zel'dovich type CMB
anisotropies.  Scalar energy density fluctuations on the surface of last
scattering induced by primordial (dark) matter density perturbations will
result both in subsequent structure formation and in appropriate dipole,
quadrupole, and higher moment anisotropies in the CMB \cite{gys}. Based on the
observed dipole asymmetry one can determine an upper limit on the expected
quadrupole anisotropy in the case of such a flat spectrum.  In addition, if
the scale of inflation is sufficiently high, long wavelength gravitational
waves will be generated during inflation whose re-entry into the horizon can
result in a large scale observed quadrupole and higher multipole anisotropies
in the CMB today \cite{gw}. While inflation is not the only method of
generating a stochastic background of horizon-sized waves \cite{krauss}, it
is certainly the most well motivated.

In this work we re-examine gravitational wave generation during inflation and
determine the predicted signal in the CMB and compare this with the COBE data.
In the process of deriving detailed estimates we update and reconcile various
earlier analyses.  We present a likelihood function for the probability that
inflation at a given scale would result in a quadrupole anisotropy at least as
big as that which is observed.  We also briefly compare this to the nature and
magnitude of the expected quadrupole anisotropy resulting from scalar density
perturbations expected from inflation.  Our analysis allows us to place both
upper and lower limits on the range of scales for which gravitational waves
from inflation could result in all or most of the observed quadrupole
anisotropy.  These scales are consistent with the scale at which the
$SU(3)\times SU(2)\times U(1)$ gauge couplings can be unified, based on a
renormalization group extrapolation of low energy data, for SUSY GUT models.
We find this coincidence both suggestive and exciting, and consider other
possible observational probes of a gravitational wave background at this level.
For this purpose we calculate the energy density stored in such a stochastic
background today.

\vskip .1in

Since the work of Starobinsky \cite{star}, it has been recognized that a
period of exponential expansion in the early universe would lead to the
production of gravitational waves.
Rubakov and collaborators \cite{Anis} were the first to use this to limit the
scale of inflation and with it the scale of Grand Unification.  Since that
time analyses designed to help more accurately compute the gravitational wave
background and compare predictions to the data have been developed
\cite{FabPol,AbbWis1,AbbSch,Sahni}. More recently the the limits on the
quadrupole anisotropy of the microwave background had improved.  It thus
seemed, even before COBE, a propitious time to re-analyze the gravitational
wave limits.  Many of the analytic techniques and results we derive have
appeared in one form or another scattered in the literature, but we have made
some effort to check, unify and reconcile the previous methods and in the
process correct any errors.  Further details can be found in \cite{white}.

It is convenient to write the metric in the $k=0$ Robertson-Walker form
\begin{equation}
  ds^2 =  R^2(\tau)\left( -d\tau^2 + d\vec{x}^2 \right)
\label{eqn:metric}
\end{equation}
where $d\tau = dt/R(t)$ is the conformal time.
In a universe which undergoes a period of exponential inflation, followed by a
radiation dominated epoch and then a matter dominated phase, $R(\tau)$ and
$\dot{R}(\tau)$ can be matched at the transition points, assuming that the
transitions between phases are sudden  (this approximation is sufficient for
our purposes) \cite{AbbHar}.  Because of the matching conditions $\tau$ is
discontinuous across the transitions.  We define $\tau_1$ to be the (conformal)
time of radiation-matter equality, $\tau_2$ to be the end of inflation.
The Hubble constant during inflation, $H$, and vacuum energy density $V_0$
driving the inflation are related by
\begin{equation}
  H^2 = {8\pi\over 3} {V_0\over m_{Pl}^2} = {8\pi\over 3} m_{Pl}^2 v
\label{eqn:hubble-v0}
\end{equation}
where we introduce the notation $v\equiv V_0/m_{Pl}^4$.

A classical gravitational wave in the linearized theory is a ripple on the
background space-time
\begin{equation}
  g_{\mu\nu} = R^2(\tau) \left( \eta_{\mu\nu} + h_{\mu\nu} \right)
  \ \ \mbox{where }\ \eta_{\mu\nu}=\mbox{diag}(-1,1,1,1),
  \ \ h_{\mu\nu} \ll 1
\label{eqn:wave-metric}
\end{equation}
In transverse traceless (TT) gauge the two independent polarization states of
the wave are denoted as $+,\times$.  In the linear theory the TT metric
fluctuations are gauge invariant (they can be related to components of the
curvature tensor).  Write
\begin{equation}
  h_{\mu\nu}(\tau,\vec{x}) = h_{\lambda}(\tau;\vec{k})
  e^{i\vec{k}\cdot\vec{x}} \epsilon_{\mu\nu}(\vec{k};\lambda)
\label{eqn:plane-wave}
\end{equation}
where $\epsilon_{\mu\nu}(\vec{k};\lambda)$ is the polarization tensor and
$\lambda=+,\times$.  The equation for the amplitude
$h_{\lambda}(\tau;\vec{k})$ is obtained by requiring the perturbed metric
(\ref{eqn:wave-metric}) satisfy Einstein's equations to $O(h)$.
As was first noted by Grishchuk \cite{Grishchuk} the equation of motion for
this amplitude is then identical to the massless Klein-Gordon equation for a
plane wave in the background space-time.  In this way, one finds each
polarization state of the wave behaves as a massless, minimally coupled, real
scalar field, with a normalization factor of $\sqrt{16\pi G}$ relating the two.

The spectrum of gravitational waves generated by quantum fluctuations during
the inflationary period can be derived by a sequence of Bogoluibov
transformations relating creation and annihilation operators defined in the
various phases: inflationary, radiation and matter dominated
\cite{AbbHar,Sahni}.  The key idea is that for long wavelength (c.f. the
horizon size) modes the transitions between the phases are sudden and the
universe will remain in the quantum state it occupied before the transition
(treating each of the transitions as instantaneous is a good approximation for
all but the highest frequency graviton modes).  However the creation and
annihilation operators that describe the particles in the state are related
by a Bogoluibov transformation, so the quantum expectation value of any
string of fields is changed.
A calculation of the quantum $n$-point functions suffices to find the spectrum
of classical gravitational waves today since the statistical average of the
ensemble of classical waves can be related to the corresponding quantum
average.

A stochastic spectrum of classical gravitational waves (in terms
of comoving wavenumber $\vec{k}$) in the expanding universe
 has the form
\begin{equation}
  h_{\lambda}(\tau ;\vec{k}) = A(k) a_{\lambda}(\vec{k})
  \left[{3j_1(k\tau)\over k\tau}\right]\ \ \ \mbox{with}\ \ \lambda=+,\times
\label{eqn:spectral-amp}
\end{equation}
where the term $[\cdots]$ is a real solution of the Klein-Gordon
equation in a matter dominated universe.  $a_{\lambda}(\vec{k})$ is a random
variable with statistical expectation value (normalized to simplify the
relation between $A(k)$ and the energy density per logarithmic frequency
interval as we later describe)  \begin{equation} \left\langle
a_{\lambda}(\vec{k}) a_{\lambda'}(\vec{q}) \right\rangle = k^{-3}
\delta^{(3)}(\vec{k}-\vec{q}) \delta_{\lambda\lambda'} \label{eqn:rv-norm}
\end{equation}

Waves which are still well outside the horizon at the time of matter-radiation
equality $(k\tau_1\ll 2\pi)$ will give the largest contribution to the CMB
anisotropy today. Calculating the Bogoluibov coefficients by matching the
field and its first derivative at $\tau_2,\tau_1$ in the limit $k\tau\ll 2\pi$
one can derive the prediction for the ($k$-independent) spectrum of
long-wavelength gravitational waves generated by inflation
\cite{gw,white}
\begin{equation}
A^2(k) = {H^2\over\pi^2 m_{Pl}^2} = {8\over 3\pi}v
\label{eqn:amplitude}
\end{equation}

To make contact with observations one must consider the effect such a
spectrum will have on the CMBR.
If one expands the CMBR temperature anisotropy in spherical harmonics
\begin{equation}
{\delta T\over T}(\theta,\phi) = \sum_{lm} a_{lm} Y_{lm}(\theta,\phi)
\label{eqn:mode-expansion}
\end{equation}
one can present the prediction of a given spectrum of gravitational waves
in terms of the $a_{lm}$.
The temperature fluctuation due to a gravitational wave $h_{\mu\nu}$ can be
found using the Sachs-Wolfe formula
\begin{equation}
  {\delta T\over T} = -{1\over 2} \int_e^r d\Lambda
  \ {\partial h_{\mu\nu}(\tau,\vec{x})\over \partial\tau}
  \hat{x}^{\mu} \hat{x}^{\nu}
\label{eqn:sachs-wolfe}
\end{equation}
where $\Lambda$ is a parameter along the unperturbed path and the lower (upper)
limit of integration represents the point of emission (reception) of the
photon.

It is standard to project out a multipole and calculate the rotationally
symmetric quantity
\begin{equation}
  \left\langle a_l^2 \right\rangle \equiv
  \left\langle \sum_m |a_{lm}|^2 \right\rangle
\end{equation}
After some algebra, utilizing identities for spherical polynomials and Bessel
functions (i.e. see \cite{white}), one finds for waves entering the horizon
during the matter dominated era (the results are very insensitive to this
restriction since the $k$-integral is dominated by waves entering the horizon
today: $k\approx 2\pi /\tau_0$)
\begin{equation}
\left\langle a_l^2 \right\rangle = 36\pi^2 (2l+1){(l+2)!\over (l-2)!}
\int_0^{2\pi/\tau_1} k dk\ A^2(k) | F_l(k) |^2
\label{eqn:moment}
\end{equation}
where the function $F_l(k)$ is defined as ($\tau(r)=\tau_0-r$)
\begin{equation}
F_l(k) \equiv
\int_0^{\tau_0-\tau_1} dr \left({d\over d(k\tau)}{j_1(k\tau)\over k\tau}\right)
\left[ {j_{l-2}(kr)\over(2l-1)(2l+1)}+{2j_l(kr)\over(2l-1)(2l+3)}
      +{j_{l+2}(kr)\over(2l+1)(2l+3)} \right]
\end{equation}
Accounting for the factor of two difference between definitions of $A^2(k)$
this agrees with the result of \cite{AbbWis1}, and differs by $\approx 2$
with the earlier result of \cite{FabPol}.

The calculation of the expectation value $\langle a_l^2 \rangle$ is not the
end of the story however. One must also consider the statistical properties of
$a_l^2$ \cite{FabPol,AbbWis2,bond,white}.  Given that each of the $a_{lm}$ are
independent Gaussian random variables the probability distribution for each
$a_l^2$, with mean $\langle a_l^2\rangle$, is of the $\chi^2$ form. One can
calculate the confidence levels for $a_l^2$ in terms of the incomplete gamma
function. We find for the for the quadrupole, ${\langle a_2^2\rangle}=7.74v$
and $a_2^2/\langle a_2^2\rangle =$ .63,.32,.23 at the 68, 90 and 95\% (lower)
confidence levels respectively.


The new COBE observations can be summarized for our purposes as as a value for
the rms quadrupole moment.  If one fits to a Harrison-Zel'dovich spectrum the
quoted value is \cite{smoot}
\begin{equation}
Q_{rms-PS} \equiv \left({ a_2^2 \over 4\pi}\right)^{1/2} =
{(16.7\pm 4)\mu K\over 2.73K} \Rightarrow  a_2^2 = (4.7\pm 2) \times 10^{-10}.
\label{eqn:qrms}
\end{equation}
Note that the quoted error on $Q_{rms-PS}$ is Gaussian, while the distribution
of $a_2^2$ is $\chi^2$.  This implies that in proceeding from the inferred
value of $a_2^2$ to a value of $v$ we must be careful to properly take into
account the resultant statistics which will be far from Gaussian. In particular
the mode of the distribution will be lower than the mean (as is noted in
\cite{smoot}).  From (\ref{eqn:qrms}) the quadrupole moment is consistent with
gravitational waves resulting from a mean value of $v = 6.1 \times 10^{-11}$.
To determine the uncertainty on $v$ we have performed a simple Monte-Carlo
analysis to find the distribution for $v$ (see figure 1).
Based on this we can determine both upper and lower limits on the value of $v$
consistent with the observations and also find the most probable value of $v$.
We find
\begin{equation}
\begin{array}{cccccc}
 3.7 \times 10^{-10} & > & v & > & 2.5 \times 10^{-12} & 95\%\  CL \\
 1.5 \times 10^{-10} & > & v & > & 2.3 \times 10^{-11} & 68\%\  CL
\end{array}
\label{eqn:limits}
\end{equation}
with a maximum likelihood value of $v \approx 4 \times 10^{-11}$.

These limits as quoted require some interpretation.  First the 95\% upper
limit $v<3.7\times 10^{-10}$ provides a strict upper limit on the scale of
inflation $\approx (v^{1/4}M_{Pl})=5.2\times 10^{16}$GeV assuming that the
contribution to the quadrupole moment from scalar density perturbations is
insignificant.  This could be increased slightly in the unlikely case that a
comparable quadrupole moment from density perturbations existed and happened
to cancel out that due to gravitational waves to some degree.

In this regard it is worthwhile considering what magnitude of quadrupole moment
is expected from scalar density perturbations from inflation. By requiring
that the induced dipole due to long wavelength modes not greatly exceed the
observed dipole anisotropy one can put an upper limit on the overall magnitude
of a flat spectrum of perturbations at horizon crossing and from this an upper
limit on all higher multipoles. At the 90\% confidence  level an upper limit
of $a_2^2\approx 2\times 10^{-10}$ has been derived \cite{AbbSch}.
Equivalently, fitting observed clustering to a primordial fluctuations spectrum
\cite{bond} one can predict a value of $a_2^2$. Such an analysis yields best
fit values in the range $a_2^2\approx (1.9 -9.9)\times 10^{-11}$.
While these estimates are probably consistent with the COBE observation, they
also suggest that a major fraction of the observed anisotropy may be due to
gravitational waves.  Note that \cite{AbbWis2,AbbSch} both the gravitational
wave induced anisotropy and the density fluctuation induced anisotropy yield
similar distributions of moments at least up to $l\approx 10$.  Thus
measurements of the correlation function cannot easily serve to distinguish
between these possibilities at this stage.

What scale of inflation, M, in GeV, do the above limits correspond to?  From
(\ref{eqn:limits}) we find, at the 95\% confidence level,
$1.5\times 10^{16}\mbox{GeV}<M<5.2\times 10^{16}$GeV, with the best fit value
$2.9 \times 10^{16}$GeV.
On the other hand using data from precision electroweak measurements at LEP on
the strong and weak coupling constants one finds, for minimal $SU(5)$ SUSY
models with SUSY breaking between $M_Z$ and $1$TeV, that coupling constant
unification can occur at a single GUT scale,$M_X$ in the range
$M_X \approx (1-3.6) \times 10^{16}$GeV \cite{wilczek}.  Unfortunately there
are no explicit compelling SUSY or GUT inflationary scenarios with which one
can compare, but generically, unless there is fine tuning, or hierarchies, in
a GUT scenario  $V_0\approx \kappa M^4$ where $\kappa \approx .01-1$ (for
example in a Coleman-Weinberg $SU(5)$ model $\kappa=9/32\pi^2$).
Thus the energy scale of inflation consistent with the observed quadrupole
anisotropy coming from gravitational waves can coincide with the estimated
scale of SUSY Grand Unification as inferred from extrapolation of low energy
couplings.  We find this possibility both plausible and exciting.  At the very
least it is quite promising that COBE through the quadrupole anisotropy is
sensitive to gravitational waves from inflation at interesting scales.

Since both density perturbations and gravitational wave anisotropies resulting
from inflation result in a flat spectrum for the CMB anisotropy, with a great
similarity in all multipoles up to at least $l=9$, it will be difficult from
CMB measurements alone to verify whether or not the observed signal is due
gravitational waves.   How might one hope then to distinguish between these
possibilities?  The simplest way would be to probe for evidence of a flat
spectrum of gravitational waves in regions of smaller wavenumber.   At present,
the most sensitive gravitational wave detector at shorter wavelengths (periods
of $O$(years)) is also astrophysical in origin, and is based on timing
measurements of millisecond pulsars \cite{rees,kragw,taylor}.
On smaller wavelengths still terrestrial probes, such as the proposed LIGO
gravity wave detector \cite{thorne}, are currently envisaged.

The sensitivity of all such detectors is based on the mean energy density per
logarithmic frequency interval in gravitational waves.  For waves which come
inside the horizon during the matter dominated era we can utilize
(\ref{eqn:spectral-amp}) and (\ref{eqn:rv-norm}).  Averaging over many
wavelengths/periods, summing over helicities,  and also taking the stochastic
average, we find
\begin{equation}
  k {d\rho_g\over dk} = {k_{phys}^2\over 2G}
  \ A^2(k)\overline{\left[{3j_1(k\tau)\over k\tau}\right]^2}
\end{equation}
where $k_{phys}=k/R(\tau)$.  For wavelengths much smaller than the horizon
($k\tau\gg 2\pi$) this goes as $R^{-4}(\tau)$ as expected.  The time evolution
factor $\overline{(3j_1(k\tau)/k\tau)^2}$ is crucial, and in fact implies that
the energy density in gravitational waves also redshifts considerably as it
comes inside the horizon.  Thus the energy density at horizon crossing is
considerably smaller than the asymptotic value when the wave is well outside
the horizon, a fact which has not been stressed before to our knowledge.
In any case dividing by the critical density today we find for waves just
coming inside the horizon,
\begin{equation}
(\Omega_g)_{hc}\approx \left\{ \begin{array}{cccc}
 16v/9    &=& {2/3\pi}   ({H_{infl}/ {M_{Pl}}})^2   & RD \\
  v/\pi^2 &=& {3/8\pi^2} ({H_{infl}/ {M_{Pl}}})^2   & MD
\end{array} \right.
\end{equation}
The result at horizon crossing in a radiation dominated epoch was calculated
using the appropriate Bogolubov coefficients.  This results in the factor of
$3j_1(k\tau)/ k\tau$ above being changed to $j_0(k\tau)$. Waves which come
inside the horizon during the radiation dominated era will redshift with one
extra power of $R$ compared to matter during the matter dominated era.
Thus their contribution to $\Omega$ today will be suppressed compared to their
contribution at horizon crossing by the factor
$\rho_{rad}/\rho_c=4\times 10^{-5} h^{-2}$,
where the Hubble constant today is $100h$km/sec/Mpc. As a result, we find
that such waves today, taking $v<3.7\times 10^{-10}$, form a stochastic
background with $\Omega_g<2.6\times 10^{-14}h^{-2}<10^{-13};\ (h>0.5)$.

The waves for which the millisecond pulsar timing and future interferometer
measurements are sensitive entered the horizon during the radiation dominated
era.  The present limit, at the 68{\%} confidence level, from pulsar timing
data is $\Omega_g<9\times 10^{-8}$ \cite{taylor}.  This limit can improve in
principle with the measuring time to the fourth power \cite{rees,kragw} but,
even in the most optimistic case, dedicated observations with many pulsars,
and better clocks, over a period of perhaps a century would be required to
uncover such a signal.  The expected energy density is also about 2 orders of
magnitude below the optimum projected capabilities of future terrestrial
detectors.

Thus, prospects look grim, without some significant technological advances, for
detecting such a gravitational wave background directly anywhere but in the
microwave background signal.  Barring a very refined measurement of high
multipoles in the CMB anisotropy we may have to await confirmation at
accelerators, or perhaps proton decay detectors, before we can determine
whether COBE has discovered the first evidence for GUTs, supersymmetry, and
at the very least, gravitational waves.

\vskip .1in

\noindent We thank Mark Wise and Vince Moncrief for very helpful discussions.


\newpage

\begin{thebibliography}{99}
\bibitem{smoot}G.F. Smoot {\it et al}, COBE preprint, April 21, 1992,
submitted to  Astrophys. J.
\bibitem{penzias}A.A. Penzias and R.W. Wilson, Astrophys. J.
{\bf 142} (1965) 419
\bibitem{gys} A. Guth and S.-Y. Pi, Phys. Rev. Lett. {\bf 49} (1982) 1110;
S. Hawking, Phys. Lett. {\bf B115} (1982) 295;
A.A. Starobinsky, Phys. Lett. {\bf B117} (1982) 175;
J. Bardeen, P. Steinhardt and M. Turner, Phys. Rev. {\bf D28} (1983) 679;
L.F. Abbott and M.B. Wise, Ap. J. {\bf 282} (1984) L47;
J.R. Bond and G. Efstathiou, Mon. Not. R. Astr. Soc. {\bf 226} (1987) 655
\bibitem{gw} V.A. Rubakov, M.V. Sazhin and A.V. Veryaskin,
Phys. Lett. {\bf B115} (1982) 189;
L.F. Abbott and M.B. Wise, Nucl. Phys. {\bf B244} (1984) 541
\bibitem{krauss} L.M. Krauss, {\it Gravitational Waves from Global Phase
Transitions}, Phys. Lett. B, in press.
\bibitem{star}A.A. Starobinsky, JETP Lett. {\bf 30} (1979) 682;
\bibitem{Anis}V.A. Rubakov, M.V. Sazhin and A.V. Veryaskin, Phys. Lett.
{\bf B115} (1982) 189; A.A. Starobinsky, Sov. Astron. Lett. {\bf 9} (1983) 302;
A.A. Starobinsky, Sov. Astron. Lett. {\bf 11} (1985) 133
\bibitem{FabPol} R. Fabbri and M.D. Pollock, Phys. Lett. {\bf B125} (1983) 445;
 R. Fabbri in Proceedings of the International School of Physics, Enrico Fermi,
1982, course 86, ed. F. Melchiorri and R. Ruffini (North-Holland)
\bibitem{AbbWis1} L.F. Abbott and M.B. Wise, Nucl. Phys. {\bf B244} (1984) 541
\bibitem{AbbSch} L.F. Abbott and R. Schaefer, Ap.J. {\bf 308} (1986) 546
\bibitem{Sahni} V. Sahni, Phys. Rev. {\bf D42} (1990) 453
\bibitem{white}  M. White, to appear
\bibitem{AbbHar}L.F. Abbott and D.D. Harari, Nucl. Phys. {\bf B264} (1986) 487
\bibitem{Grishchuk} L.P. Grishchuk, Sov.Phys. JETP {\bf 40} (1975) 409;
L.P. Grishchuk, Ann. N.Y. Acad. Sci. {\bf 302} (1977) 439
\bibitem{AbbWis2} L.F. Abbott and M.B. Wise, Ap. J. {\bf 282} (1984) L47;
\bibitem{bond} J.R. Bond and G. Efstathiou, Mon. Not. R. Astr. Soc.
{\bf 226} (1987) 655
\bibitem{wilczek} see S.Dimopoulos, S.A. Raby, F. Wilczek, Phys. Today,
Oct. 1991, for early references;
U. Amaldi {\it et al}, Phys. Lett. {\bf B 135} (1991);
P. Langacker, M. Luo, U. Penn. preprint (1991)
\bibitem{rees} B. Bertotti, B.J. Carr, M.J. Rees, Mon. Not. R. Ast. Soc.
{\bf 203} (1983) 945
\bibitem{kragw} L.M. Krauss, Nature {\bf 313} (1985) 32
\bibitem{taylor} i.e. see D. R. Stinebring {\it et al},
Phys. Rev. Lett. {\bf 65} (1990) 285
\bibitem{thorne}  see K.S. Thorne, in {\it 300 Years of Gravitation},
ed. by S. W. Hawking and W. Israel (Cambridge Univ. Press, Cambridge, 1989)
\end{thebibliography}

\newpage

\noindent{\bf{Figure Captions}}

\vskip .1in

\noindent Figure 1:  The distribution for the scale of inflation
$v=(V_0/M_Pl^{4})$ as determined by Monte Carlo, using the COBE measurements
and assuming the observed quadrupole anisotropy is due to gravitational waves.

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%  Figure 1       -   Post script.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
/s {stroke} def /l {lineto} def /m {moveto} def /t {translate} def
/sw {stringwidth} def /r {rotate} def /rl {roll} def
/d {rlineto} def /rm {rmoveto} def /gr {grestore} def /f {eofill} def
/c {setrgbcolor} def /lw {setlinewidth} def /sd {setdash} def
/mp {newpath /y exch def /x exch def} def
/side {w 0 l currentpoint t -144 r} def
/mr {mp x y w2 0 360 arc} def /m24 {mr s} def /m20 {mr f} def
/mb {mp x y w2 add m w2 neg 0 d 0 w neg d w 0 d 0 wd cl} def
/mt {mp x y w2 add m w2 neg w neg d w 0 d cl} def
/m21 {mb f} def /m25 {mb s} def /m22 {mt f} def /m26{mt s} def
/m23 {mp x y w2 sub m w2 w d w neg 0 d cl f} def
 /m27 {mp x y w2 add m w3 neg w2 neg d w3 w2 neg d w3 w2 d cl s} def
 /m28 {mp x w2 sub y w2 sub w3 add m w3 0 d 0 w3 neg d w3 0 d 0 w3 d w3 0 d
 0 w3 d w3 neg 0 d 0 w3 d w3 neg 0 d 0 w3 neg d w3 neg 0 d cl s } def
 /m29 {mp gsave x w2 sub y w2 add w3 sub m currentpoint t
 4 {side} repeat cl fill gr} def
 /m30 {mp gsave x w2 sub y w2 add w3 sub m currentpoint t
 4 {side} repeat cl s gr} def /m31 {mp x y w2 sub m 0 w d x w2 sub y m w 0 d
 x w2 sub y w2 add m w w neg d x w2 sub y w2
 sub m w w d s} def
/m2 {mp x y w2 sub m 0 w d x w2 sub y m w 0 d} def
/m5 {mp x w2 sub y w2 sub m w w d x w2 sub y w2 add m w w neg d} def
/cl {closepath} def /sf {scalefont setfont} def
/box {m dup 0 exch d exch 0 d 0 exch neg d cl} def
/bl {box s} def /bf {box f} def
/oshow {gsave [] 0 sd true charpath stroke gr} def
/p {gsave 4 2 rl t exch r dup sw pop neg} def
/rs {p 0 m show gr} def /ors {p 0 m oshow gr} def
/cs {p 2 div 0 m show gr} def /ocs {p 2 div 0 m oshow gr} def
/reencsmalldict 24 dict def /ReEncodeSmall
{reencsmalldict begin /newcodesandnames exch def /newfontname exch def
/basefontname exch def /basefontdict basefontname findfont def
 /newfont basefontdict maxlength dict def basefontdict
{exch dup /FID ne {dup /Encoding eq {exch dup length array copy
newfont 3 1 roll put} {exch newfont 3 1 roll put} ifelse}
{pop pop} ifelse } forall
newfont /FontName newfontname put newcodesandnames aload pop
newcodesandnames length 2 idiv {newfont /Encoding get 3 1 roll put} repeat
newfontname newfont definefont pop end } def /accvec [
8#260 /agrave 8#265 /Agrave 8#276 /acircumflex 8#300 /Acircumflex
8#311 /adieresis 8#314 /Adieresis 8#321 /ccedilla 8#322 /Ccedilla
8#323 /eacute 8#324 /Eacute 8#325 /egrave 8#326 /Egrave
8#327 /ecircumflex 8#330 /Ecircumflex 8#331 /edieresis 8#332 /Edieresis
8#333 /icircumflex 8#334 /Icircumflex 8#335 /idieresis 8#336 /Idieresis
8#337 /ntilde 8#340 /Ntilde 8#342 /ocircumflex 8#344 /Ocircumflex
8#345 /odieresis 8#346 /Odieresis 8#347 /ucircumflex 8#354 /Ucircumflex
8#355 /udieresis 8#356 /Udieresis 8#357 /aring 8#362 /Aring
8#363 /ydieresis 8#364 /Ydieresis 8#366 /aacute 8#367 /Aacute
8#374 /ugrave 8#375 /Ugrave] def
/Times-Roman /Times-Roman accvec ReEncodeSmall
/Times-Italic /Times-Italic accvec ReEncodeSmall
/Times-Bold /Times-Bold accvec ReEncodeSmall
/Times-BoldItalic /Times-BoldItalic accvec ReEncodeSmall
/Helvetica /Helvetica accvec ReEncodeSmall
/Helvetica-Oblique /Helvetica-Oblique accvec ReEncodeSmall
/Helvetica-Bold /Helvetica-Bold accvec ReEncodeSmall
/Helvetica-BoldOblique /Helvetica-BoldOblique  accvec ReEncodeSmall
/Courier /Courier accvec ReEncodeSmall
/Courier-Oblique /Courier-Oblique accvec ReEncodeSmall
/Courier-Bold /Courier-Bold accvec ReEncodeSmall
/Courier-BoldOblique /Courier-BoldOblique accvec ReEncodeSmall
/Zone {/iy exch def /ix exch def gsave ix 1 sub 2076 mul 1 iy sub 2848
 mul t} def
%%EndProlog
gsave 40 90 t .25 .25 scale gsave
 1 1 Zone
 gsave 0 0 t 0 0 0 c [] 0 sd 1 lw 1558 1662 312 593 bl 312 593 m 312 2255 l s
 343 593 m 312 593 l s 327 626 m 312 626 l s 327 659 m 312 659 l s 327 693 m
312
 693 l s 327 726 m 312 726 l s 343 759 m 312 759 l s 327 792 m 312 792 l s 327
 826 m 312 826 l s 327 859 m 312 859 l s 327 892 m 312 892 l s 343 925 m 312
925
 l s 327 959 m 312 959 l s 327 992 m 312 992 l s 327 1025 m 312 1025 l s 327
 1058 m 312 1058 l s 343 1092 m 312 1092 l s 327 1125 m 312 1125 l s 327 1158 m
 312 1158 l s 327 1191 m 312 1191 l s 327 1225 m 312 1225 l s 343 1258 m 312
 1258 l s 327 1291 m 312 1291 l s 327 1324 m 312 1324 l s 327 1358 m 312 1358 l
 s 327 1391 m 312 1391 l s 343 1424 m 312 1424 l s 327 1457 m 312 1457 l s 327
 1490 m 312 1490 l s 327 1524 m 312 1524 l s 327 1557 m 312 1557 l s 343 1590 m
 312 1590 l s 327 1623 m 312 1623 l s 327 1657 m 312 1657 l s 327 1690 m 312
 1690 l s 327 1723 m 312 1723 l s 343 1756 m 312 1756 l s 327 1790 m 312 1790 l
 s 327 1823 m 312 1823 l s 327 1856 m 312 1856 l s 327 1889 m 312 1889 l s 343
 1923 m 312 1923 l s 327 1956 m 312 1956 l s 327 1989 m 312 1989 l s 327 2022 m
 312 2022 l s 327 2056 m 312 2056 l s 343 2089 m 312 2089 l s 327 2122 m 312
 2122 l s 327 2155 m 312 2155 l s 327 2189 m 312 2189 l s 327 2222 m 312 2222 l
 s 343 2255 m 312 2255 l s 255 607 m 251 606 l 248 602 l 247 595 l 247 591 l
248
 584 l 251 580 l 255 578 l 258 578 l 262 580 l 265 584 l 266 591 l 266 595 l
265
 602 l 262 606 l 258 607 l 255 607 l cl s 186 774 m 181 772 l 179 768 l 177 761
 l 177 757 l 179 750 l 181 746 l 186 745 l 188 745 l 193 746 l 195 750 l 197
757
 l 197 761 l 195 768 l 193 772 l 188 774 l 186 774 l cl s 208 747 m 206 746 l
 208 745 l 209 746 l 208 747 l cl s 220 767 m 220 768 l 222 771 l 223 772 l 226
 774 l 231 774 l 234 772 l 235 771 l 237 768 l 237 765 l 235 763 l 233 758 l
219
 745 l 238 745 l s 263 774 m 249 774 l 248 761 l 249 763 l 253 764 l 258 764 l
 262 763 l 265 760 l 266 756 l 266 753 l 265 749 l 262 746 l 258 745 l 253 745
l
 249 746 l 248 747 l 247 750 l s 213 940 m 209 939 l 206 934 l 205 927 l 205
923
 l 206 916 l 209 912 l 213 911 l 216 911 l 220 912 l 223 916 l 224 923 l 224
927
 l 223 934 l 220 939 l 216 940 l 213 940 l cl s 235 914 m 234 912 l 235 911 l
 237 912 l 235 914 l cl s 263 940 m 249 940 l 248 927 l 249 929 l 253 930 l 258
 930 l 262 929 l 265 926 l 266 922 l 266 919 l 265 915 l 262 912 l 258 911 l
253
 911 l 249 912 l 247 916 l s 186 1106 m 181 1105 l 179 1101 l 177 1094 l 177
 1089 l 179 1083 l 181 1078 l 186 1077 l 188 1077 l 193 1078 l 195 1083 l 197
 1089 l 197 1094 l 195 1101 l 193 1105 l 188 1106 l 186 1106 l cl s 208 1080 m
 206 1078 l 208 1077 l 209 1078 l 208 1080 l cl s 238 1106 m 224 1077 l s 219
 1106 m 238 1106 l s 263 1106 m 249 1106 l 248 1094 l 249 1095 l 253 1096 l 258
 1096 l 262 1095 l 265 1092 l 266 1088 l 266 1085 l 265 1081 l 262 1078 l 258
 1077 l 253 1077 l 249 1078 l 248 1080 l 247 1083 l s 251 1267 m 253 1268 l 258
 1272 l 258 1243 l s 181 1433 m 184 1434 l 188 1439 l 188 1409 l s 208 1412 m
 206 1411 l 208 1409 l 209 1411 l 208 1412 l cl s 220 1432 m 220 1433 l 222
1436
 l 223 1437 l 226 1439 l 231 1439 l 234 1437 l 235 1436 l 237 1433 l 237 1430 l
 235 1427 l 233 1423 l 219 1409 l 238 1409 l s 263 1439 m 249 1439 l 248 1426 l
 249 1427 l 253 1429 l 258 1429 l 262 1427 l 265 1425 l 266 1421 l 266 1418 l
 265 1414 l 262 1411 l 258 1409 l 253 1409 l 249 1411 l 248 1412 l 247 1415 l s
 209 1599 m 212 1601 l 216 1605 l 216 1576 l s 235 1578 m 234 1577 l 235 1576 l
 237 1577 l 235 1578 l cl s 263 1605 m 249 1605 l 248 1592 l 249 1594 l 253
1595
 l 258 1595 l 262 1594 l 265 1591 l 266 1587 l 266 1584 l 265 1580 l 262 1577 l
 258 1576 l 253 1576 l 249 1577 l 248 1578 l 247 1581 l s 181 1765 m 184 1767 l
 188 1771 l 188 1742 l s 208 1745 m 206 1743 l 208 1742 l 209 1743 l 208 1745 l
 cl s 238 1771 m 224 1742 l s 219 1771 m 238 1771 l s 263 1771 m 249 1771 l 248
 1759 l 249 1760 l 253 1761 l 258 1761 l 262 1760 l 265 1757 l 266 1753 l 266
 1750 l 265 1746 l 262 1743 l 258 1742 l 253 1742 l 249 1743 l 247 1747 l s 248
 1930 m 248 1932 l 249 1934 l 251 1936 l 253 1937 l 259 1937 l 262 1936 l 263
 1934 l 265 1932 l 265 1929 l 263 1926 l 260 1922 l 247 1908 l 266 1908 l s 179
 2096 m 179 2098 l 180 2101 l 181 2102 l 184 2103 l 190 2103 l 193 2102 l 194
 2101 l 195 2098 l 195 2095 l 194 2092 l 191 2088 l 177 2074 l 197 2074 l s 208
 2077 m 206 2076 l 208 2074 l 209 2076 l 208 2077 l cl s 220 2096 m 220 2098 l
 222 2101 l 223 2102 l 226 2103 l 231 2103 l 234 2102 l 235 2101 l 237 2098 l
 237 2095 l 235 2092 l 233 2088 l 219 2074 l 238 2074 l s 263 2103 m 249 2103 l
 248 2091 l 249 2092 l 253 2094 l 258 2094 l 262 2092 l 265 2090 l 266 2085 l
 266 2083 l 265 2078 l 262 2076 l 258 2074 l 253 2074 l 249 2076 l 248 2077 l
 247 2080 l s 206 2263 m 206 2264 l 208 2267 l 209 2268 l 212 2270 l 217 2270 l
 220 2268 l 222 2267 l 223 2264 l 223 2261 l 222 2259 l 219 2254 l 205 2241 l
 224 2241 l s 235 2243 m 234 2242 l 235 2241 l 237 2242 l 235 2243 l cl s 263
 2270 m 249 2270 l 248 2257 l 249 2259 l 253 2260 l 258 2260 l 262 2259 l 265
 2256 l 266 2252 l 266 2249 l 265 2245 l 262 2242 l 258 2241 l 253 2241 l 249
 2242 l 248 2243 l 247 2246 l s 312 593 m 1870 593 l s 312 624 m 312 593 l s
343
 608 m 343 593 l s 374 608 m 374 593 l s 405 608 m 405 593 l s 436 608 m 436
593
 l s 467 624 m 467 593 l s 499 608 m 499 593 l s 530 608 m 530 593 l s 561 608
m
 561 593 l s 592 608 m 592 593 l s 623 624 m 623 593 l s 654 608 m 654 593 l s
 686 608 m 686 593 l s 717 608 m 717 593 l s 748 608 m 748 593 l s 779 624 m
779
 593 l s 810 608 m 810 593 l s 841 608 m 841 593 l s 873 608 m 873 593 l s 904
 608 m 904 593 l s 935 624 m 935 593 l s 966 608 m 966 593 l s 997 608 m 997
593
 l s 1028 608 m 1028 593 l s 1060 608 m 1060 593 l s 1091 624 m 1091 593 l s
 1122 608 m 1122 593 l s 1153 608 m 1153 593 l s 1184 608 m 1184 593 l s 1215
 608 m 1215 593 l s 1247 624 m 1247 593 l s 1278 608 m 1278 593 l s 1309 608 m
 1309 593 l s 1340 608 m 1340 593 l s 1371 608 m 1371 593 l s 1402 624 m 1402
 593 l s 1434 608 m 1434 593 l s 1465 608 m 1465 593 l s 1496 608 m 1496 593 l
s
 1527 608 m 1527 593 l s 1558 624 m 1558 593 l s 1589 608 m 1589 593 l s 1621
 608 m 1621 593 l s 1652 608 m 1652 593 l s 1683 608 m 1683 593 l s 1714 624 m
 1714 593 l s 1745 608 m 1745 593 l s 1776 608 m 1776 593 l s 1808 608 m 1808
 593 l s 1839 608 m 1839 593 l s 1870 624 m 1870 593 l s 310 572 m 306 571 l
303
 567 l 302 560 l 302 556 l 303 549 l 306 544 l 310 543 l 313 543 l 317 544 l
320
 549 l 321 556 l 321 560 l 320 567 l 317 571 l 313 572 l 310 572 l cl s 445 572
 m 441 571 l 438 567 l 437 560 l 437 556 l 438 549 l 441 544 l 445 543 l 448
543
 l 452 544 l 455 549 l 456 556 l 456 560 l 455 567 l 452 571 l 448 572 l 445
572
 l cl s 467 546 m 466 544 l 467 543 l 469 544 l 467 546 l cl s 495 572 m 481
572
 l 480 560 l 481 561 l 485 562 l 490 562 l 494 561 l 497 558 l 498 554 l 498
551
 l 497 547 l 494 544 l 490 543 l 485 543 l 481 544 l 480 546 l 479 549 l s 618
 567 m 621 568 l 625 572 l 625 543 l s 753 567 m 756 568 l 760 572 l 760 543 l
s
 779 546 m 778 544 l 779 543 l 781 544 l 779 546 l cl s 807 572 m 793 572 l 792
 560 l 793 561 l 797 562 l 801 562 l 805 561 l 808 558 l 810 554 l 810 551 l
808
 547 l 805 544 l 801 543 l 797 543 l 793 544 l 792 546 l 790 549 l s 927 565 m
 927 567 l 929 571 l 932 572 l 938 572 l 941 571 l 943 567 l 943 564 l 942 561
l
 939 557 l 925 543 l 945 543 l s 1062 565 m 1062 567 l 1064 571 l 1067 572 l
 1073 572 l 1076 571 l 1078 567 l 1078 564 l 1077 561 l 1074 557 l 1060 543 l
 1080 543 l s 1091 546 m 1089 544 l 1091 543 l 1092 544 l 1091 546 l cl s 1119
 572 m 1105 572 l 1103 560 l 1105 561 l 1109 562 l 1113 562 l 1117 561 l 1120
 558 l 1121 554 l 1121 551 l 1120 547 l 1117 544 l 1113 543 l 1109 543 l 1105
 544 l 1103 546 l 1102 549 l s 1240 572 m 1255 572 l 1247 561 l 1251 561 l 1254
 560 l 1255 558 l 1256 554 l 1256 551 l 1255 547 l 1252 544 l 1248 543 l 1244
 543 l 1240 544 l 1238 546 l 1237 549 l s 1375 572 m 1390 572 l 1382 561 l 1386
 561 l 1389 560 l 1390 558 l 1391 554 l 1391 551 l 1390 547 l 1387 544 l 1383
 543 l 1379 543 l 1375 544 l 1373 546 l 1372 549 l s 1402 546 m 1401 544 l 1402
 543 l 1404 544 l 1402 546 l cl s 1430 572 m 1416 572 l 1415 560 l 1416 561 l
 1420 562 l 1425 562 l 1429 561 l 1432 558 l 1433 554 l 1433 551 l 1432 547 l
 1429 544 l 1425 543 l 1420 543 l 1416 544 l 1415 546 l 1414 549 l s 1562 572 m
 1549 553 l 1569 553 l s 1562 572 m 1562 543 l s 1698 572 m 1684 553 l 1704 553
 l s 1698 572 m 1698 543 l s 1714 546 m 1713 544 l 1714 543 l 1716 544 l 1714
 546 l cl s 1742 572 m 1728 572 l 1727 560 l 1728 561 l 1732 562 l 1736 562 l
 1740 561 l 1743 558 l 1745 554 l 1745 551 l 1743 547 l 1740 544 l 1736 543 l
 1732 543 l 1728 544 l 1727 546 l 1725 549 l s 1877 572 m 1863 572 l 1862 560 l
 1863 561 l 1867 562 l 1871 562 l 1876 561 l 1878 558 l 1880 554 l 1880 551 l
 1878 547 l 1876 544 l 1871 543 l 1867 543 l 1863 544 l 1862 546 l 1860 549 l s
 1870 593 m 1870 2255 l s 1839 593 m 1870 593 l s 1854 626 m 1870 626 l s 1854
 659 m 1870 659 l s 1854 693 m 1870 693 l s 1854 726 m 1870 726 l s 1839 759 m
 1870 759 l s 1854 792 m 1870 792 l s 1854 826 m 1870 826 l s 1854 859 m 1870
 859 l s 1854 892 m 1870 892 l s 1839 925 m 1870 925 l s 1854 959 m 1870 959 l
s
 1854 992 m 1870 992 l s 1854 1025 m 1870 1025 l s 1854 1058 m 1870 1058 l s
 1839 1092 m 1870 1092 l s 1854 1125 m 1870 1125 l s 1854 1158 m 1870 1158 l s
 1854 1191 m 1870 1191 l s 1854 1225 m 1870 1225 l s 1839 1258 m 1870 1258 l s
 1854 1291 m 1870 1291 l s 1854 1324 m 1870 1324 l s 1854 1358 m 1870 1358 l s
 1854 1391 m 1870 1391 l s 1839 1424 m 1870 1424 l s 1854 1457 m 1870 1457 l s
 1854 1490 m 1870 1490 l s 1854 1524 m 1870 1524 l s 1854 1557 m 1870 1557 l s
 1839 1590 m 1870 1590 l s 1854 1623 m 1870 1623 l s 1854 1657 m 1870 1657 l s
 1854 1690 m 1870 1690 l s 1854 1723 m 1870 1723 l s 1839 1756 m 1870 1756 l s
 1854 1790 m 1870 1790 l s 1854 1823 m 1870 1823 l s 1854 1856 m 1870 1856 l s
 1854 1889 m 1870 1889 l s 1839 1923 m 1870 1923 l s 1854 1956 m 1870 1956 l s
 1854 1989 m 1870 1989 l s 1854 2022 m 1870 2022 l s 1854 2056 m 1870 2056 l s
 1839 2089 m 1870 2089 l s 1854 2122 m 1870 2122 l s 1854 2155 m 1870 2155 l s
 1854 2189 m 1870 2189 l s 1854 2222 m 1870 2222 l s 1839 2255 m 1870 2255 l s
 312 2255 m 1870 2255 l s 312 2224 m 312 2255 l s 343 2240 m 343 2255 l s 374
 2240 m 374 2255 l s 405 2240 m 405 2255 l s 436 2240 m 436 2255 l s 467 2224 m
 467 2255 l s 499 2240 m 499 2255 l s 530 2240 m 530 2255 l s 561 2240 m 561
 2255 l s 592 2240 m 592 2255 l s 623 2224 m 623 2255 l s 654 2240 m 654 2255 l
 s 686 2240 m 686 2255 l s 717 2240 m 717 2255 l s 748 2240 m 748 2255 l s 779
 2224 m 779 2255 l s 810 2240 m 810 2255 l s 841 2240 m 841 2255 l s 873 2240 m
 873 2255 l s 904 2240 m 904 2255 l s 935 2224 m 935 2255 l s 966 2240 m 966
 2255 l s 997 2240 m 997 2255 l s 1028 2240 m 1028 2255 l s 1060 2240 m 1060
 2255 l s 1091 2224 m 1091 2255 l s 1122 2240 m 1122 2255 l s 1153 2240 m 1153
 2255 l s 1184 2240 m 1184 2255 l s 1215 2240 m 1215 2255 l s 1247 2224 m 1247
 2255 l s 1278 2240 m 1278 2255 l s 1309 2240 m 1309 2255 l s 1340 2240 m 1340
 2255 l s 1371 2240 m 1371 2255 l s 1402 2224 m 1402 2255 l s 1434 2240 m 1434
 2255 l s 1465 2240 m 1465 2255 l s 1496 2240 m 1496 2255 l s 1527 2240 m 1527
 2255 l s 1558 2224 m 1558 2255 l s 1589 2240 m 1589 2255 l s 1621 2240 m 1621
 2255 l s 1652 2240 m 1652 2255 l s 1683 2240 m 1683 2255 l s 1714 2224 m 1714
 2255 l s 1745 2240 m 1745 2255 l s 1776 2240 m 1776 2255 l s 1808 2240 m 1808
 2255 l s 1839 2240 m 1839 2255 l s 1870 2224 m 1870 2255 l s 1716 500 m 1725
 481 l s 1733 500 m 1725 481 l s 1754 500 m 1769 481 l s 1769 500 m 1754 481 l
s
 1795 504 m 1798 506 l 1802 510 l 1802 481 l s 1827 510 m 1823 508 l 1820 504 l
 1819 497 l 1819 493 l 1820 486 l 1823 482 l 1827 481 l 1830 481 l 1834 482 l
 1837 486 l 1838 493 l 1838 497 l 1837 504 l 1834 508 l 1830 510 l 1827 510 l
cl
 s 1846 514 m 1848 515 l 1850 517 l 1850 503 l s 1862 517 m 1860 516 l 1859 514
 l 1858 511 l 1858 509 l 1859 505 l 1860 503 l 1866 503 l 1867 505 l 1868 509 l
 1868 511 l 1867 514 l 1866 516 l 1864 517 l 1862 517 l cl s 96 2032 m 125 2032
 l s 96 2032 m 96 2045 l 97 2049 l 98 2050 l 101 2051 l 105 2051 l 108 2050 l
 109 2049 l 111 2045 l 111 2032 l s 105 2061 m 125 2061 l s 114 2061 m 109 2063
 l 107 2065 l 105 2068 l 105 2072 l s 90 2089 m 93 2086 l 97 2083 l 109 2079 l
 115 2079 l 122 2081 l 127 2083 l 132 2086 l 134 2089 l s 105 2096 m 125 2104 l
 s 105 2112 m 125 2104 l s 90 2119 m 93 2122 l 97 2125 l 103 2128 l 109 2129 l
 115 2129 l 122 2128 l 132 2122 l 134 2119 l s 105 2153 m 125 2168 l s 105 2168
 m 125 2153 l s 101 2194 m 100 2197 l 96 2201 l 125 2201 l s 96 2226 m 97 2222
l
 101 2219 l 108 2218 l 112 2218 l 119 2219 l 123 2222 l 125 2226 l 125 2229 l
 123 2233 l 119 2236 l 112 2237 l 108 2237 l 101 2236 l 97 2233 l 96 2229 l 96
 2226 l cl s 92 2244 m 91 2244 l 90 2245 l 89 2245 l 88 2247 l 88 2250 l 90
2252
 l 94 2252 l 103 2243 l 103 2253 l s [] 0 sd 312 593 m 312 593 l 319 593 l 319
 1165 l 327 1165 l 327 1280 l 335 1280 l 335 1393 l 343 1393 l 343 1524 l 351
 1524 l 351 1596 l 358 1596 l 358 1703 l 366 1703 l 366 1806 l 374 1806 l 374
 1925 l 382 1925 l 382 2039 l 390 2039 l 390 2044 l 397 2044 l 397 2162 l 405
 2162 l 405 2116 l 413 2116 l 413 2218 l 421 2218 l 421 2187 l 429 2187 l 429
 2163 l 436 2163 l 436 2181 l 444 2181 l 444 2136 l 452 2136 l 452 2129 l 460
 2129 l 460 2068 l 467 2068 l 467 2014 l 475 2014 l 475 1968 l 483 1968 l 483
 1954 l 491 1954 l 491 1893 l 499 1893 l 499 1817 l 506 1817 l 506 1803 l 514
 1803 l 514 1735 l 522 1735 l 522 1686 l 530 1686 l 530 1618 l 538 1618 l 538
 1589 l 545 1589 l 545 1578 l 553 1578 l 553 1503 l 561 1503 l 561 1453 l 569
 1453 l 569 1425 l 577 1425 l 577 1471 l 584 1471 l 584 1385 l 592 1385 l 592
 1332 l 600 1332 l 600 1326 l 608 1326 l 608 1330 l 616 1330 l 616 1264 l 623
 1264 l 623 1226 l 631 1226 l 631 1191 l 639 1191 l 639 1197 l 647 1197 l 647
 1139 l 654 1139 l 654 1151 l 662 1151 l 662 1114 l 670 1114 l 670 1082 l 678
 1082 l 678 1090 l 686 1090 l 686 1058 l 693 1058 l 693 1054 l 701 1054 l 701
 1019 l 709 1019 l 709 992 l 717 992 l 717 977 l 725 977 l 725 969 l 732 969 l
 732 949 l 740 949 l 740 945 l 748 945 l 748 923 l 756 923 l 756 927 l 764 927
l
 764 905 l 771 905 l 771 896 l 779 896 l 779 872 l 787 872 l 787 885 l 795 885
l
 795 853 l 803 853 l 803 854 l 810 854 l 810 849 l 818 849 l 818 859 l 826 859
l
 826 799 l 834 799 l 834 834 l 841 834 l 841 827 l 849 827 l 849 834 l 857 834
l
 857 806 l 865 806 l 865 796 l 873 796 l 873 766 l 880 766 l 880 759 l 888 759
l
 888 794 l 896 794 l 896 802 l 904 802 l 904 762 l 912 762 l 912 755 l 919 755
l
 919 744 l 927 744 l 927 753 l 935 753 l 935 729 l 943 729 l 943 749 l 951 749
l
 951 744 l 958 744 l 958 752 l 966 752 l 966 721 l 974 721 l 974 719 l 982 719
l
 982 724 l 990 724 l 990 694 l 997 694 l 997 709 l 1005 709 l 1005 694 l 1013
 694 l 1013 697 l 1021 697 l 1021 691 l 1028 691 l 1028 697 l 1036 697 l 1036
 702 l 1044 702 l 1044 697 l 1052 697 l 1052 688 l 1060 688 l 1060 687 l 1067
 687 l 1067 685 l 1075 685 l 1075 672 l 1083 672 l 1083 683 l 1091 683 l 1091
 668 l 1099 668 l s 1099 668 m 1099 671 l 1122 671 l 1122 668 l 1130 668 l 1130
 675 l 1138 675 l 1138 657 l 1145 657 l 1145 661 l 1153 661 l 1153 663 l 1161
 663 l 1161 664 l 1169 664 l 1169 659 l 1177 659 l 1177 661 l 1184 661 l 1184
 648 l 1192 648 l 1192 645 l 1200 645 l 1200 655 l 1208 655 l 1208 643 l 1215
 643 l 1215 651 l 1223 651 l 1223 645 l 1231 645 l 1231 649 l 1239 649 l 1239
 631 l 1247 631 l 1247 645 l 1254 645 l 1254 639 l 1262 639 l 1262 647 l 1270
 647 l 1270 643 l 1278 643 l 1278 640 l 1286 640 l 1286 645 l 1293 645 l 1293
 638 l 1301 638 l 1301 639 l 1309 639 l 1309 633 l 1325 633 l 1325 627 l 1332
 627 l 1332 629 l 1340 629 l 1340 635 l 1348 635 l 1348 637 l 1356 637 l 1356
 639 l 1364 639 l 1364 623 l 1371 623 l 1371 635 l 1379 635 l 1379 633 l 1387
 633 l 1387 625 l 1395 625 l 1395 623 l 1402 623 l 1402 628 l 1410 628 l 1410
 627 l 1418 627 l 1418 616 l 1426 616 l 1426 618 l 1434 618 l 1434 623 l 1441
 623 l 1441 620 l 1449 620 l 1449 619 l 1457 619 l 1457 623 l 1465 623 l 1465
 632 l 1473 632 l 1473 616 l 1480 616 l 1480 619 l 1488 619 l 1488 618 l 1496
 618 l 1496 623 l 1512 623 l 1512 615 l 1519 615 l 1519 618 l 1527 618 l 1527
 621 l 1535 621 l 1535 612 l 1543 612 l 1543 610 l 1551 610 l 1551 619 l 1558
 619 l 1558 612 l 1566 612 l 1566 614 l 1574 614 l 1574 610 l 1582 610 l 1582
 613 l 1589 613 l 1589 610 l 1597 610 l 1597 609 l 1605 609 l 1605 608 l 1613
 608 l 1613 612 l 1621 612 l 1621 616 l 1628 616 l 1628 608 l 1636 608 l 1636
 609 l 1644 609 l 1644 614 l 1652 614 l 1652 608 l 1660 608 l 1660 603 l 1667
 603 l 1667 612 l 1675 612 l 1675 606 l 1683 606 l 1683 602 l 1691 602 l 1691
 612 l 1699 612 l 1699 604 l 1706 604 l 1706 615 l 1714 615 l 1714 606 l 1722
 606 l 1722 607 l 1730 607 l 1730 602 l 1738 602 l 1738 617 l 1745 617 l 1745
 614 l 1753 614 l 1753 611 l 1761 611 l 1761 606 l 1769 606 l 1769 603 l 1776
 603 l 1776 608 l 1784 608 l 1784 604 l 1800 604 l 1800 607 l 1815 607 l 1815
 606 l 1823 606 l 1823 604 l 1831 604 l 1831 606 l 1839 606 l 1839 602 l 1847
 602 l 1847 605 l 1854 605 l 1854 608 l 1862 608 l 1862 593 l 1870 593 l s
 [] 0 sd 312 593 m 312 593 l 319 593 l 319 1165 l 327 1165 l 327 1280 l 335
1280
 l 335 1393 l 343 1393 l 343 1524 l 351 1524 l 351 1596 l 358 1596 l 358 1703 l
 366 1703 l 366 1806 l 374 1806 l 374 1925 l 382 1925 l 382 2039 l 390 2039 l
 390 2044 l 397 2044 l 397 2162 l 405 2162 l 405 2116 l 413 2116 l 413 2218 l
 421 2218 l 421 2187 l 429 2187 l 429 2163 l 436 2163 l 436 2181 l 444 2181 l
 444 2136 l 452 2136 l 452 2129 l 460 2129 l 460 2068 l 467 2068 l 467 2014 l
 475 2014 l 475 1968 l 483 1968 l 483 1954 l 491 1954 l 491 1893 l 499 1893 l
 499 1817 l 506 1817 l 506 1803 l 514 1803 l 514 1735 l 522 1735 l 522 1686 l
 530 1686 l 530 1618 l 538 1618 l 538 1589 l 545 1589 l 545 1578 l 553 1578 l
 553 1503 l 561 1503 l 561 1453 l 569 1453 l 569 1425 l 577 1425 l 577 1471 l
 584 1471 l 584 1385 l 592 1385 l 592 1332 l 600 1332 l 600 1326 l 608 1326 l
 608 1330 l 616 1330 l 616 1264 l 623 1264 l 623 1226 l 631 1226 l 631 1191 l
 639 1191 l 639 1197 l 647 1197 l 647 1139 l 654 1139 l 654 1151 l 662 1151 l
 662 1114 l 670 1114 l 670 1082 l 678 1082 l 678 1090 l 686 1090 l 686 1058 l
 693 1058 l 693 1054 l 701 1054 l 701 1019 l 709 1019 l 709 992 l 717 992 l 717
 977 l 725 977 l 725 969 l 732 969 l 732 949 l 740 949 l 740 945 l 748 945 l
748
 923 l 756 923 l 756 927 l 764 927 l 764 905 l 771 905 l 771 896 l 779 896 l
779
 872 l 787 872 l 787 885 l 795 885 l 795 853 l 803 853 l 803 854 l 810 854 l
810
 849 l 818 849 l 818 859 l 826 859 l 826 799 l 834 799 l 834 834 l 841 834 l
841
 827 l 849 827 l 849 834 l 857 834 l 857 806 l 865 806 l 865 796 l 873 796 l
873
 766 l 880 766 l 880 759 l 888 759 l 888 794 l 896 794 l 896 802 l 904 802 l
904
 762 l 912 762 l 912 755 l 919 755 l 919 744 l 927 744 l 927 753 l 935 753 l
935
 729 l 943 729 l 943 749 l 951 749 l 951 744 l 958 744 l 958 752 l 966 752 l
966
 721 l 974 721 l 974 719 l 982 719 l 982 724 l 990 724 l 990 694 l 997 694 l
997
 709 l 1005 709 l 1005 694 l 1013 694 l 1013 697 l 1021 697 l 1021 691 l 1028
 691 l 1028 697 l 1036 697 l 1036 702 l 1044 702 l 1044 697 l 1052 697 l 1052
 688 l 1060 688 l 1060 687 l 1067 687 l 1067 685 l 1075 685 l 1075 672 l 1083
 672 l 1083 683 l 1091 683 l 1091 668 l 1099 668 l s 1099 668 m 1099 671 l 1122
 671 l 1122 668 l 1130 668 l 1130 675 l 1138 675 l 1138 657 l 1145 657 l 1145
 661 l 1153 661 l 1153 663 l 1161 663 l 1161 664 l 1169 664 l 1169 659 l 1177
 659 l 1177 661 l 1184 661 l 1184 648 l 1192 648 l 1192 645 l 1200 645 l 1200
 655 l 1208 655 l 1208 643 l 1215 643 l 1215 651 l 1223 651 l 1223 645 l 1231
 645 l 1231 649 l 1239 649 l 1239 631 l 1247 631 l 1247 645 l 1254 645 l 1254
 639 l 1262 639 l 1262 647 l 1270 647 l 1270 643 l 1278 643 l 1278 640 l 1286
 640 l 1286 645 l 1293 645 l 1293 638 l 1301 638 l 1301 639 l 1309 639 l 1309
 633 l 1325 633 l 1325 627 l 1332 627 l 1332 629 l 1340 629 l 1340 635 l 1348
 635 l 1348 637 l 1356 637 l 1356 639 l 1364 639 l 1364 623 l 1371 623 l 1371
 635 l 1379 635 l 1379 633 l 1387 633 l 1387 625 l 1395 625 l 1395 623 l 1402
 623 l 1402 628 l 1410 628 l 1410 627 l 1418 627 l 1418 616 l 1426 616 l 1426
 618 l 1434 618 l 1434 623 l 1441 623 l 1441 620 l 1449 620 l 1449 619 l 1457
 619 l 1457 623 l 1465 623 l 1465 632 l 1473 632 l 1473 616 l 1480 616 l 1480
 619 l 1488 619 l 1488 618 l 1496 618 l 1496 623 l 1512 623 l 1512 615 l 1519
 615 l 1519 618 l 1527 618 l 1527 621 l 1535 621 l 1535 612 l 1543 612 l 1543
 610 l 1551 610 l 1551 619 l 1558 619 l 1558 612 l 1566 612 l 1566 614 l 1574
 614 l 1574 610 l 1582 610 l 1582 613 l 1589 613 l 1589 610 l 1597 610 l 1597
 609 l 1605 609 l 1605 608 l 1613 608 l 1613 612 l 1621 612 l 1621 616 l 1628
 616 l 1628 608 l 1636 608 l 1636 609 l 1644 609 l 1644 614 l 1652 614 l 1652
 608 l 1660 608 l 1660 603 l 1667 603 l 1667 612 l 1675 612 l 1675 606 l 1683
 606 l 1683 602 l 1691 602 l 1691 612 l 1699 612 l 1699 604 l 1706 604 l 1706
 615 l 1714 615 l 1714 606 l 1722 606 l 1722 607 l 1730 607 l 1730 602 l 1738
 602 l 1738 617 l 1745 617 l 1745 614 l 1753 614 l 1753 611 l 1761 611 l 1761
 606 l 1769 606 l 1769 603 l 1776 603 l 1776 608 l 1784 608 l 1784 604 l 1800
 604 l 1800 607 l 1815 607 l 1815 606 l 1823 606 l 1823 604 l 1831 604 l 1831
 606 l 1839 606 l 1839 602 l 1847 602 l 1847 605 l 1854 605 l 1854 608 l 1862
 608 l 1862 593 l 1870 593 l s [12 12] 0 sd 312 593 m 312 593 l 319 593 l 319
 1165 l 327 1165 l 327 1280 l 335 1280 l 335 1393 l 343 1393 l 343 1524 l 351
 1524 l 351 1596 l 358 1596 l 358 1703 l 366 1703 l 366 1806 l 374 1806 l 374
 1925 l 382 1925 l 382 2039 l 390 2039 l 390 2044 l 397 2044 l 397 2162 l 405
 2162 l 405 2116 l 413 2116 l 413 2218 l 421 2218 l 421 2187 l 429 2187 l 429
 2163 l 436 2163 l 436 2181 l 444 2181 l 444 2136 l 452 2136 l 452 2129 l 460
 2129 l 460 2068 l 467 2068 l 467 2014 l 475 2014 l 475 1968 l 483 1968 l 483
 1954 l 491 1954 l 491 1893 l 499 1893 l 499 1817 l 506 1817 l 506 1803 l 514
 1803 l 514 1735 l 522 1735 l 522 1686 l 530 1686 l 530 1618 l 538 1618 l 538
 1589 l 545 1589 l 545 1578 l 553 1578 l 553 1503 l 561 1503 l 561 1453 l 569
 1453 l 569 1425 l 577 1425 l 577 1471 l 584 1471 l 584 1385 l 592 1385 l 592
 1332 l 600 1332 l 600 1326 l 608 1326 l 608 1330 l 616 1330 l 616 1264 l 623
 1264 l 623 1226 l 631 1226 l 631 1191 l 639 1191 l 639 1197 l 647 1197 l 647
 1139 l 654 1139 l 654 1151 l 662 1151 l 662 1114 l 670 1114 l 670 1082 l 678
 1082 l 678 1090 l 686 1090 l 686 1058 l 693 1058 l 693 1054 l 701 1054 l 701
 1019 l 709 1019 l 709 992 l 717 992 l 717 977 l 725 977 l 725 969 l 732 969 l
 732 949 l 740 949 l 740 945 l 748 945 l 748 923 l 756 923 l 756 927 l 764 927
l
 764 905 l 771 905 l 771 896 l 779 896 l 779 872 l 787 872 l 787 885 l 795 885
l
 795 853 l 803 853 l 803 854 l 810 854 l 810 849 l 818 849 l 818 859 l 826 859
l
 826 799 l 834 799 l 834 834 l 841 834 l 841 827 l 849 827 l 849 834 l 857 834
l
 857 806 l 865 806 l 865 796 l 873 796 l 873 766 l 880 766 l 880 759 l 888 759
l
 888 794 l 896 794 l 896 802 l 904 802 l 904 762 l 912 762 l 912 755 l 919 755
l
 919 744 l 927 744 l 927 753 l 935 753 l 935 729 l 943 729 l 943 749 l 951 749
l
 951 744 l 958 744 l 958 752 l 966 752 l 966 721 l 974 721 l 974 719 l 982 719
l
 982 724 l 990 724 l 990 694 l 997 694 l 997 709 l 1005 709 l 1005 694 l 1013
 694 l 1013 697 l 1021 697 l 1021 691 l 1028 691 l 1028 697 l 1036 697 l 1036
 702 l 1044 702 l 1044 697 l 1052 697 l 1052 688 l 1060 688 l 1060 687 l 1067
 687 l 1067 685 l 1075 685 l 1075 672 l 1083 672 l 1083 683 l 1091 683 l 1091
 668 l 1099 668 l s 1099 668 m 1099 671 l 1122 671 l 1122 668 l 1130 668 l 1130
 675 l 1138 675 l 1138 657 l 1145 657 l 1145 661 l 1153 661 l 1153 663 l 1161
 663 l 1161 664 l 1169 664 l 1169 659 l 1177 659 l 1177 661 l 1184 661 l 1184
 648 l 1192 648 l 1192 645 l 1200 645 l 1200 655 l 1208 655 l 1208 643 l 1215
 643 l 1215 651 l 1223 651 l 1223 645 l 1231 645 l 1231 649 l 1239 649 l 1239
 631 l 1247 631 l 1247 645 l 1254 645 l 1254 639 l 1262 639 l 1262 647 l 1270
 647 l 1270 643 l 1278 643 l 1278 640 l 1286 640 l 1286 645 l 1293 645 l 1293
 638 l 1301 638 l 1301 639 l 1309 639 l 1309 633 l 1325 633 l 1325 627 l 1332
 627 l 1332 629 l 1340 629 l 1340 635 l 1348 635 l 1348 637 l 1356 637 l 1356
 639 l 1364 639 l 1364 623 l 1371 623 l 1371 635 l 1379 635 l 1379 633 l 1387
 633 l 1387 625 l 1395 625 l 1395 623 l 1402 623 l 1402 628 l 1410 628 l 1410
 627 l 1418 627 l 1418 616 l 1426 616 l 1426 618 l 1434 618 l 1434 623 l 1441
 623 l 1441 620 l 1449 620 l 1449 619 l 1457 619 l 1457 623 l 1465 623 l 1465
 632 l 1473 632 l 1473 616 l 1480 616 l 1480 619 l 1488 619 l 1488 618 l 1496
 618 l 1496 623 l 1512 623 l 1512 615 l 1519 615 l 1519 618 l 1527 618 l 1527
 621 l 1535 621 l 1535 612 l 1543 612 l 1543 610 l 1551 610 l 1551 619 l 1558
 619 l 1558 612 l 1566 612 l 1566 614 l 1574 614 l 1574 610 l 1582 610 l 1582
 613 l 1589 613 l 1589 610 l 1597 610 l 1597 609 l 1605 609 l 1605 608 l 1613
 608 l 1613 612 l 1621 612 l 1621 616 l 1628 616 l 1628 608 l 1636 608 l 1636
 609 l 1644 609 l 1644 614 l 1652 614 l 1652 608 l 1660 608 l 1660 603 l 1667
 603 l 1667 612 l 1675 612 l 1675 606 l 1683 606 l 1683 602 l 1691 602 l 1691
 612 l 1699 612 l 1699 604 l 1706 604 l 1706 615 l 1714 615 l 1714 606 l 1722
 606 l 1722 607 l 1730 607 l 1730 602 l 1738 602 l 1738 617 l 1745 617 l 1745
 614 l 1753 614 l 1753 611 l 1761 611 l 1761 606 l 1769 606 l 1769 603 l 1776
 603 l 1776 608 l 1784 608 l 1784 604 l 1800 604 l 1800 607 l 1815 607 l 1815
 606 l 1823 606 l 1823 604 l 1831 604 l 1831 606 l 1839 606 l 1839 602 l 1847
 602 l 1847 605 l 1854 605 l 1854 608 l 1862 608 l 1862 593 l 1870 593 l s
 [4 8] 0 sd 312 593 m 312 593 l 319 593 l 319 1165 l 327 1165 l 327 1280 l 335
 1280 l 335 1393 l 343 1393 l 343 1524 l 351 1524 l 351 1596 l 358 1596 l 358
 1703 l 366 1703 l 366 1806 l 374 1806 l 374 1925 l 382 1925 l 382 2039 l 390
 2039 l 390 2044 l 397 2044 l 397 2162 l 405 2162 l 405 2116 l 413 2116 l 413
 2218 l 421 2218 l 421 2187 l 429 2187 l 429 2163 l 436 2163 l 436 2181 l 444
 2181 l 444 2136 l 452 2136 l 452 2129 l 460 2129 l 460 2068 l 467 2068 l 467
 2014 l 475 2014 l 475 1968 l 483 1968 l 483 1954 l 491 1954 l 491 1893 l 499
 1893 l 499 1817 l 506 1817 l 506 1803 l 514 1803 l 514 1735 l 522 1735 l 522
 1686 l 530 1686 l 530 1618 l 538 1618 l 538 1589 l 545 1589 l 545 1578 l 553
 1578 l 553 1503 l 561 1503 l 561 1453 l 569 1453 l 569 1425 l 577 1425 l 577
 1471 l 584 1471 l 584 1385 l 592 1385 l 592 1332 l 600 1332 l 600 1326 l 608
 1326 l 608 1330 l 616 1330 l 616 1264 l 623 1264 l 623 1226 l 631 1226 l 631
 1191 l 639 1191 l 639 1197 l 647 1197 l 647 1139 l 654 1139 l 654 1151 l 662
 1151 l 662 1114 l 670 1114 l 670 1082 l 678 1082 l 678 1090 l 686 1090 l 686
 1058 l 693 1058 l 693 1054 l 701 1054 l 701 1019 l 709 1019 l 709 992 l 717
992
 l 717 977 l 725 977 l 725 969 l 732 969 l 732 949 l 740 949 l 740 945 l 748
945
 l 748 923 l 756 923 l 756 927 l 764 927 l 764 905 l 771 905 l 771 896 l 779
896
 l 779 872 l 787 872 l 787 885 l 795 885 l 795 853 l 803 853 l 803 854 l 810
854
 l 810 849 l 818 849 l 818 859 l 826 859 l 826 799 l 834 799 l 834 834 l 841
834
 l 841 827 l 849 827 l 849 834 l 857 834 l 857 806 l 865 806 l 865 796 l 873
796
 l 873 766 l 880 766 l 880 759 l 888 759 l 888 794 l 896 794 l 896 802 l 904
802
 l 904 762 l 912 762 l 912 755 l 919 755 l 919 744 l 927 744 l 927 753 l 935
753
 l 935 729 l 943 729 l 943 749 l 951 749 l 951 744 l 958 744 l 958 752 l 966
752
 l 966 721 l 974 721 l 974 719 l 982 719 l 982 724 l 990 724 l 990 694 l 997
694
 l 997 709 l 1005 709 l 1005 694 l 1013 694 l 1013 697 l 1021 697 l 1021 691 l
 1028 691 l 1028 697 l 1036 697 l 1036 702 l 1044 702 l 1044 697 l 1052 697 l
 1052 688 l 1060 688 l 1060 687 l 1067 687 l 1067 685 l 1075 685 l 1075 672 l
 1083 672 l 1083 683 l 1091 683 l 1091 668 l 1099 668 l s 1099 668 m 1099 671 l
 1122 671 l 1122 668 l 1130 668 l 1130 675 l 1138 675 l 1138 657 l 1145 657 l
 1145 661 l 1153 661 l 1153 663 l 1161 663 l 1161 664 l 1169 664 l 1169 659 l
 1177 659 l 1177 661 l 1184 661 l 1184 648 l 1192 648 l 1192 645 l 1200 645 l
 1200 655 l 1208 655 l 1208 643 l 1215 643 l 1215 651 l 1223 651 l 1223 645 l
 1231 645 l 1231 649 l 1239 649 l 1239 631 l 1247 631 l 1247 645 l 1254 645 l
 1254 639 l 1262 639 l 1262 647 l 1270 647 l 1270 643 l 1278 643 l 1278 640 l
 1286 640 l 1286 645 l 1293 645 l 1293 638 l 1301 638 l 1301 639 l 1309 639 l
 1309 633 l 1325 633 l 1325 627 l 1332 627 l 1332 629 l 1340 629 l 1340 635 l
 1348 635 l 1348 637 l 1356 637 l 1356 639 l 1364 639 l 1364 623 l 1371 623 l
 1371 635 l 1379 635 l 1379 633 l 1387 633 l 1387 625 l 1395 625 l 1395 623 l
 1402 623 l 1402 628 l 1410 628 l 1410 627 l 1418 627 l 1418 616 l 1426 616 l
 1426 618 l 1434 618 l 1434 623 l 1441 623 l 1441 620 l 1449 620 l 1449 619 l
 1457 619 l 1457 623 l 1465 623 l 1465 632 l 1473 632 l 1473 616 l 1480 616 l
 1480 619 l 1488 619 l 1488 618 l 1496 618 l 1496 623 l 1512 623 l 1512 615 l
 1519 615 l 1519 618 l 1527 618 l 1527 621 l 1535 621 l 1535 612 l 1543 612 l
 1543 610 l 1551 610 l 1551 619 l 1558 619 l 1558 612 l 1566 612 l 1566 614 l
 1574 614 l 1574 610 l 1582 610 l 1582 613 l 1589 613 l 1589 610 l 1597 610 l
 1597 609 l 1605 609 l 1605 608 l 1613 608 l 1613 612 l 1621 612 l 1621 616 l
 1628 616 l 1628 608 l 1636 608 l 1636 609 l 1644 609 l 1644 614 l 1652 614 l
 1652 608 l 1660 608 l 1660 603 l 1667 603 l 1667 612 l 1675 612 l 1675 606 l
 1683 606 l 1683 602 l 1691 602 l 1691 612 l 1699 612 l 1699 604 l 1706 604 l
 1706 615 l 1714 615 l 1714 606 l 1722 606 l 1722 607 l 1730 607 l 1730 602 l
 1738 602 l 1738 617 l 1745 617 l 1745 614 l 1753 614 l 1753 611 l 1761 611 l
 1761 606 l 1769 606 l 1769 603 l 1776 603 l 1776 608 l 1784 608 l 1784 604 l
 1800 604 l 1800 607 l 1815 607 l 1815 606 l 1823 606 l 1823 604 l 1831 604 l
 1831 606 l 1839 606 l 1839 602 l 1847 602 l 1847 605 l 1854 605 l 1854 608 l
 1862 608 l 1862 593 l 1870 593 l s
gr gr gr showpage gr
%%%%%%%%%%%%%%%%%%%%%%%%%%
%% End of figure 1
%%%%%%%%%%%%%%%%%%%%%%%%%%

