E8_MSalem_renorm.jpg (JPEG Image, 969x969 pixels) - Scaled (86%)

[Roger Bagula]

Loading Image...
I just ran a renormalization Julia of this E_11 polynomial against
McMullen's Salem
and I got a symmetry that has an angular characteristic much like the
big bang one?
Here is a real mouthful:
[x_] := ((x* (-1 + 22 x - 221 x^2 + 1360 x^3 - 5820 x^4 + 18664 x^5 -
47147 x^6 + 97094 x^7 - 166899 x^8 + 243304 x^9 - 303891 x^10 +
327070 x^11 - 303891 x^12 + 243304 x^13 - 166899 x^14 + 97094 x^15 -
47147 x^16 + 18664 x^17 - 5820 x^18 + 1360 x^19 - 221 x^20 + 22 x^21 -
x^22))/(x^22 + x^21 - x^19 - 2*x^18 - 3*x^17 - 3*x^16 - 2*x^15 + 2*x^13
+ 4* x^12 + 5*x^11 + 4*x^10 + 2*x^9 - 2*x^7 - 3*x^6 - 3*x^5 - 2*x^4 -
x^3 + x + 1))^2/Sqrt[2]

That E_11 breaking symmetry to give an K3 like tetrahedral symmetry.
The ratio of the McMullen Salem is close to an 1/alpha factor too.
The K3 idea was "hot" because it is related to Calabi-Yau/ null Ricci
type mirror symmetry ideas, too. McMullen plotted Siegel disk
on the terahedral surface in his paper ( cited in the sequence below).
E_11 is the symmetry above E_8 that is associated with an Hyperbolic
geometry
and has a finite algebra. Most people have been looking to an hyperbolic
E_8
group as being big bang related:
as being better than the other candidates of the Thurston and Weeks spaces.
It isn't a perfect model, but it seems to make at least some sense in a
physics way?
The only reason I knew about this Salem polynomial was I was interested
in the Siegel disks
and the tetraheral surface Implicit in McMullen's article. It isn't
real clear to me how he derived it,
but it sure works nice.

Here are sequences involving the polynomials:

%I A000001
%S A000001 -1, -22, -263, -2284, -16225, -100490, -564096, -2943434,
-14525316,
-68623698, -313160381, -1389603972, -6026265844, -25641735564,
-107383041717,
-443700335414, -1812509085585, -7331932395596, -29409752732192,
-117108140185676, -463355891177610, -1823137506023896, -7138294557783177,
-27828525782153482, -108074098915295326, -418286873399478030,
-1614019671638138278, -6211054492352336922, -23843284734435360211,
-91331025030620873572, -349153576077251792179
%N A000001 Coefficient expansion of the McMullen transformed characteristic
polynomial of the E_11 Cartan Matix:
p(x)]=-2 - 167 x + 1694 x^2 - 6069 x^3 + 11210 x^4 - 12297 x^5 + 8554
x^6 -3875
x^7 + 1140 x^8 - 210 x^9 + 22 x^10 - x^11.
%C A000001 The polynomial is symmetric:
h[x]=x^22*h[1/x].
It would be fairly easy to map the McMullem 22nd degree
Salem which is related to K3 dynamics (and a tetrahedral surface) to
this 22nd
degree polynomial. The McMullen Salem polynomial has the ratio of
1.3728862806447408 which is near 1/(100*Alpha) the fine structure constyant.
%F A000001 p(x)]=-2 - 167 x + 1694 x^2 - 6069 x^3 + 11210 x^4 - 12297
x^5 + 8554
x^6 -3875 x^7 + 1140 x^8 - 210 x^9 + 22 x^10 - x^11;
q(x)=x^11*p(x+1/x);
a(n)=Coefficient_Expansion(q(x)).
%t A000001 m11 = {{2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{-1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0},
{0, -1, 2, -1, 0, 0, 0, 0, 0, 0, -1},
{0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0},
{0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0},
{0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0},
{0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0},
{0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0},
{0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0},
{0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0},
{0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 2}};
f[x_] = CharacteristicPolynomial[m11, x];
h[x_] = ExpandAll[x^11*f[x + 1/x]];
g[x] = ExpandAll[x^22*h[1/x]];
a = Table[SeriesCoefficient[Series[1/g[x], {x, 0, 30}], n], {n, 0, 30}]
%O A000001 1
%K A000001 ,nonn,
%A A000001 Roger L. Bagula and Gary W. Adamson (***@yahoo.com),
Oct 24
2008
RH
RA 192.20.225.32
RU
RI


%I A000001
%S A000001 1, -1, 12, -12, 91, -89, 560, -526, 3061, -2715, 15526,
-12779, 74893,
-56092, 348808, -232184, 1584273, -909357, 7065982, -3354913, 31100725,
-11473678, 135587365, -34883109, 587116592, -82703752, 2530527727,
-52581912,
10874166572, 1107267567, 46648306254
%N A000001 A real root polynomial relate to McMullen's Salem gives
the polynomial used to get this expansion sequence:
p(x)=x11 + x10 - 11*x9 - 11*x8 + 42*x7 + 40*x6 - 66*x5 - 54*x4 + 42*x3 +
24*x2 - 8*x - 1.
%C A000001 McMullen's transform gives:
x11*p(x+1/x)=x22 + x21 - x19 - 2*x18 - 3*x17 - 3*x16 - 2*x15 + 2*x13 +
4*x12 + 5*x11 + 4*x10 + 2*x9 - 2*x7 - 3*x6 - 3*x5 - 2*x4 - x3 + x + 1.
%F A000001 p(x)=x11 + x10 - 11*x9 - 11*x8 + 42*x7 + 40*x6 - 66*x5 -
54*x4 + 42*x3 + 24*x2 - 8*x - 1;
a(n)=Coefficient_Expansion(x20*p(1/x)).
%t A000001 f[x_]=x11 + x10 - 11*x9 - 11*x8 + 42*x7 + 40*x6 - 66*x5 -
54*x4 + 42*x3 + 24*x2 - 8*x - 1;
g[x] = ExpandAll[x10*f[1/x]];
a = Table[SeriesCoefficient[Series[1/g[x], {x, 0, 30}], n], {n, 0, 30}]
%O A000001 1
%K A000001 ,nonn,
%A A000001 Roger L. Bagula and Gary W. Adamson (***@yahoo.com),
Oct 24 2008
RH
RA 192.20.225.32
RU
RI
%I A000001
%S A000001 1, -1, 1, 0, 1, 1, 1, 2, 2, 4, 4, 7, 9, 12, 17, 23, 32, 44,
60, 83, 113, 156,
214, 294, 403, 554, 760, 1044, 1433, 1967, 2701
%N A000001 Coefficient expansion of Salem degree 22 polynomial:
p(x)=x22 + x21 - x19 - 2*x18 - 3*x17 - 3*x16 - 2*x15 + 2*x13 + 4*x12 +
5*x11 + 4*x10 + 2*x9 - 2*x7 - 3*x6 - 3*x5 - 2*x4 - x3 + x + 1.
%D A000001 Curtis T. McMullen,Dynamics on K3 surfaces: Salem numbers and
Siegel disks,2005,
http://abel.math.harvard.edu/~ctm/papers/index.html
%F A000001 p(x)=x22 + x21 - x19 - 2*x18 - 3*x17 - 3*x16 - 2*x15 + 2*x13
+ 4*x12 + 5*x11 + 4*x10 + 2*x9 - 2*x7 - 3*x6 - 3*x5 - 2*x4 - x3 + x + 1;
a(n)=Coefficient_Expansion(p(x)).
%t A000001 f[x_] = x22 + x21 - x19 - 2*x18 - 3*x17 - 3*x16 - 2*x15 +
2*x13 + 4*x12 + 5*x11 + 4*x10 + 2*x9 - 2*x7 - 3*x6 - 3*x5 - 2*x4 - x3 +
x + 1;
g[x] = ExpandAll[x22*f[1/x]];
a = Table[SeriesCoefficient[Series[1/g[x], {x, 0, 30}], n], {n, 0, 30}]
%O A000001 1
%K A000001 ,nonn,
%A A000001 Roger L. Bagula and Gary W. Adamson (***@yahoo.com),
Oct 23 2008
RH
RA 192.20.225.32
RU
RI

Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: [phone]
:http://www.geocities.com/rlbagulatftn/Index.html
alternative email: ***@sbcglobal.net

Mathematica:
Clear[nz, f, g, arraym, j]
f[x_] := ((x* (-1 + 22 x -
221 x^2 + 1360 x^3 - 5820 x^4 + 18664 x^5 - 47147
x^6 + 97094 x^7 - 166899 x^8 + 243304 x^9 -
303891 x^10 + 327070
x^11 - 303891 x^12 + 243304 x^13 - 166899 x^14 +
97094 x^15 - 47147 x^16 + 18664 x^17 - 5820 x^18 +
1360 x^19 - 221 x^20 + 22 x^21 - x^22))/(x^22 + x^21 -
x^19 - 2*x^18 - 3*x^17 - 3*x^16 - 2*x^15 + 2*x^13 + 4*
x^12 + 5*x^11 + 4*x^10 + 2*x^9 - 2*x^7 - 3*x^6 - 3*x^5 -
2*x^4 - x^3 + x + 1))^2/Sqrt[2]
(*3D Mandelbrot of
E_8 Polynomial transformed against McMullen's 22nd
degree K3 Salem polynomial associated with a
tetrahedral surface*)
(*Julia with SQRT(x^2 + y^2) limited measure*)
(*by R. L. BAGULA 22 Oct 2008 © *)
numberOfz2ToEscape[z_] := Block[
{escapeCount, nz = N[z], nzold = 0},
For[
escapeCount = 0,
(Sqrt[Re[nz]^2 + Im[nz]^2] < 32) && (escapeCount < 255) &&
(Abs[nz -
nzold] > 10^(-3)),
nzold = nz;
nz = f[nz];
++escapeCount
];
escapeCount
]
FractalPureM[{{ReMin_, ReMax_, ReSteps_},
{ImMin_, ImMax_, ImSteps_}}] :=
Table[
numberOfz2ToEscape[x + y I],
{y, ImMin, ImMax, (ImMax - ImMin)/ImSteps},
{x, ReMin, ReMax, (ReMax - ReMin)/ReSteps}

]

arraym = FractalPureM[{{-1*Pi, 7*Pi, 100}, {-4*Pi, 4*Pi, 100}}];

gr = ListPlot3D[arraym,
Mesh -> False,
AspectRatio -> Automatic, Boxed -> False, Axes
-> False];

[Roger Bagula]

Loading Image...
Self-similar 8 legged turtles...
Pre-McMullen transform polynomial renormalized to the characteristic
polynomial of the E_11 Cartan Matrix:
f[x_] := ((x* (-2 - 167 x + 1694 x2 - 6069 x3 + 11210 x4 - 12297 x5 +
8554 x6 - 3875 x7 + 1140 x8 - 210 x9 + 22 x10 - x11))/(x11 + x10 - 11*x9
- 11*x8 + 42*x7 + 40*x6 - 66*x5 - 54*x4 + 42*x3 + 24*x2 - 8*x - 1))2/Sqrt[2]

The exploding early universe would be self0similar due to the field
renormalization
associated with the symmetry breaking that takes place during cooling.
Phase changes in the early expansion stages of the big bang.
E_11 is Planck black body-like
Flux=sigma*T4
to four U(1)*SU(11) or A_10 groups as temperature groups.
( at about 6.4*10^(-18) seconds after the big bang?)
SU(11) would be pre the SU(5) level of mass formation in the Higgs model
of mass and the stadard model cooling to the present U(1)*SU(2)*SU(3)
of the standard model of physics.
That makes at least three symmetry breaking steps during the cooling process
and probably inflationary periods associated with those phase changes.

There is some humor in this result?
http://www.nescb.org/epublications/october2000/septeditorial.html

The earth, some legends say, is borne on the back of a giant turtle.

Throughout history, turtles have been revered in art, mythology, and
customs of diverse cultures around the world.

[Knecht]

Post by Roger Bagula
http://www.geocities.com/rlbagulatftn/e_11premcsalemrenorm.jpg
Self-similar 8 legged turtles...
Pre-McMullen transform polynomial renormalized to the characteristic
f[x_] := ((x* (-2 - 167 x + 1694 x2 - 6069 x3 + 11210 x4 - 12297 x5 +
8554 x6 - 3875 x7 + 1140 x8 - 210 x9 + 22 x10 - x11))/(x11 + x10 - 11*x9
- 11*x8 + 42*x7 + 40*x6 - 66*x5 - 54*x4 + 42*x3 + 24*x2 - 8*x - 1))2/Sqrt[2]
The exploding early universe would be self0similar due to the field
renormalization
associated with the symmetry breaking that takes place during cooling.
Phase changes in the early expansion stages of the big bang.
E_11 is Planck black body-like
Flux=sigma*T4
to four U(1)*SU(11) or A_10 groups as temperature groups.
( at about 6.4*10^(-18) seconds after the big bang?)
SU(11) would be pre the SU(5)  level of mass formation in the Higgs model
of mass and the stadard model cooling to the present U(1)*SU(2)*SU(3)
of the standard model of physics.
That makes at least three symmetry breaking steps during the cooling process
and probably inflationary periods associated with those phase changes.
There is some humor in this result?http://www.nescb.org/epublications/october2000/septeditorial.html
 > The earth, some legends say, is borne on the back of a giant turtle.
Throughout history, turtles have been revered in art, mythology, and
customs of diverse cultures around the world.

Can you make any definitive predictions regarding this model, i.e.,
predictions that are unique, quantitative and testable? Or is it just
more abstract arm-waving? Could excessive analytical dithering impede
scientific progress by distracting us from studying the real world of
nature?

Yours in science,
Knecht
www.amherst.edu/~rloldershaw

[Rev. 11D Meow!]

weird how when nobody responds for days
and pseudo-3rd-party steps into view.

Then ALL HELL BREAKS LOOSE!


Let the Nobel Prize go to the spoiler!

Post by Roger Bagula
http://www.geocities.com/rlbagulatftn/e_11premcsalemrenorm.jpg
Self-similar 8 legged turtles...
Pre-McMullen transform polynomial renormalized to the characteristic
f[x_] := ((x* (-2 - 167 x + 1694 x2 - 6069 x3 + 11210 x4 - 12297 x5 +
8554 x6 - 3875 x7 + 1140 x8 - 210 x9 + 22 x10 - x11))/(x11 + x10 - 11*x9
- 11*x8 + 42*x7 + 40*x6 - 66*x5 - 54*x4 + 42*x3 + 24*x2 - 8*x - 1))2/Sqrt[2]
The exploding early universe would be self0similar due to the field
renormalization
associated with the symmetry breaking that takes place during cooling.
Phase changes in the early expansion stages of the big bang.
E_11 is Planck black body-like
Flux=sigma*T4
to four U(1)*SU(11) or A_10 groups as temperature groups.
( at about 6.4*10^(-18) seconds after the big bang?)
SU(11) would be pre the SU(5) level of mass formation in the Higgs model
of mass and the stadard model cooling to the present U(1)*SU(2)*SU(3)
of the standard model of physics.
That makes at least three symmetry breaking steps during the cooling process
and probably inflationary periods associated with those phase changes.
There is some humor in this
result?http://www.nescb.org/epublications/october2000/septeditorial.html

The earth, some legends say, is borne on the back of a giant turtle.

Throughout history, turtles have been revered in art, mythology, and
customs of diverse cultures around the world.

Can you make any definitive predictions regarding this model, i.e.,
predictions that are unique, quantitative and testable? Or is it just
more abstract arm-waving? Could excessive analytical dithering impede
scientific progress by distracting us from studying the real world of
nature?

Yours in science,
Knecht
www.amherst.edu/~rloldershaw

[Roger Bagula]

Post by Rev. 11D Meow!
weird how when nobody responds for days
and pseudo-3rd-party steps into view.
Then ALL HELL BREAKS LOOSE!
Let the Nobel Prize go to the spoiler!

To the newbie troll:
you aren't amusing.

[Knecht]

Post by Roger Bagula

Post by Rev. 11D Meow!
weird how when nobody responds for days
and pseudo-3rd-party steps into view.
Then ALL HELL BREAKS LOOSE!
Let the Nobel Prize go to the spoiler!

you aren't amusing.

Oh, but you certainly are!

[Roger Bagula]

Post by Knecht
Can you make any definitive predictions regarding this model, i.e.,
predictions that are unique, quantitative and testable? Or is it just
more abstract arm-waving? Could excessive analytical dithering impede
scientific progress by distracting us from studying the real world of
nature?
Yours in science,
Knecht
www.amherst.edu/~rloldershaw

http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation
The angular results of the E_11 Cartan polynomial
match the power spectrum of the cosmic microwave background radiation
temperature anisotropy.
It isn't exact, but it is better than E_8 which
is Janna Levin's favorite
in " How the Universe got it's Spots".
Weeks space and Thurston space are the other well known candidates.
Mathematica:
Clear[p, x]
m11 = {{2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{-1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0},
{0, -1, 2, -1, 0, 0, 0, 0, 0, 0, -1},
{0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0},
{0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0},
{0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0},
{0, 0, 0,0, 0, -1,2, -1, 0, 0, 0},
{0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0},
{0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0},
{0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0},
{0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 2}};
p[x_] = (CharacteristicPolynomial[m11, x]);
Plot[p[x], {x, 0, 6}]
Loading Image...

[Roger Bagula]

http://en.wikipedia.org/wiki/Weeks_manifold
The volume of the Weeks space manifold is:
V_weeks=0.9427
Using the Borel form volume on E_11 as characteristed by the Cartan
Matrix polynomial:
M = {{2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{-1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0},
{0, -1, 2, -1, 0, 0, 0, 0, 0, 0, -1},
{0, 0, -1, 2, -1, 0, 0, 0,0, 0, 0},
{0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0},
{0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0},
{0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0},
{0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0},
{0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0},
{0, 0, 0, 0, 0, 0, 0, 0, -1,2, 0},
{0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 2}};
f[x_] = CharacteristicPolynomial[M, x];
g[x_] = x^11*f[1/x];
a = -Table[SeriesCoefficient[
Series[1/g[x], {x, 0, 50}], n], {n, 0, 50}];
zetak[s_] := Sum[1/a[[n]]^s, {n, 1, Length[a]}];
N[3 *23^{3/2}zetak[2]/(4*Pi^4) ]
This approimation of the Zeta to fifty terms is good sine it converges
rapidly:
V_E_11=0.851051
I'm really not sure about this:
I just pulged into the formulas.
But that makes the E_11 hyperbolic manifold
somewhat smaller than Weeks space
and also gives a potental ( spring force)
for the explosive big bang expansion due to the symmetry breaking
to an higher volume hyperbolic manifold.
The universe, right now, is supposed to be nearly a flat Euclidean 3
manifold.

[Roger Bagula]

The Zeta function I used was wrong,
but no good Dedekind zeta for a Caratn E_11 polynomial based number
field actually exists in Mathematica as far as I can find:
http://www.google.com/codesearch?hl=en&q=Dedekind+zeta+lang:mathematica+show:pgCL-zjjl-o:HOwEw8zylow:1Dz7fuo7zas&sa=N&cd=1&ct=rc&cs_p=http://mcs.open.ac.uk&cs_f=ugg2/coincidences/MultipleCoincidences.nb
Gives a cyclotomic number field version....
Hurwitz generralized version of Zeta is slightly smaller volume for
Weeks space:
N[3*23^(3/2)Zeta[2, -x /. NSolve[x^3 - x + 1 == 0,
x][[1]]]/(4*Pi^4)]=0.938754
Doing that for the Cartan E_11 Polynomial:
N[3*23^(3/2)Zeta[2, x /. NSolve[f[x] == 0, x][[11]]]/(4*Pi^4)]=0.240318
Which is way too low!
Generally the way the volume to curvature realtionship goes is:
3 Manifold 3d volume formula:
V=(4*Pi*r3/3)*( 1+k*R02/r2)
k=-1 for hyperbolic negative curvature
k=0 Euclidean zero curvature
k=1 for sphere elliptic plus curvature.,
R0 is the Manifold reference curvature number.
In cosmology this is like the Robertson-Walker metric:
ds^2=c^2-R(t)^2*(dr^2/(1-k*r^2)+r^2*dtheta^2+r^2*Sin[theta]^2*dPhi^2)
(In Einstein and Riemann you see a factor of r^2/4).
In other words in an elliptical space the volume is more than a flate
Euclidean
and in a hyperbolic it is less.
The R0 factor is manifold or symmetry group dependant.
The Weeks space has the highest known R0 for a hyperbolic space.
The higher the R0 number the lower the volume
and the more vacuum energy the space has over a WEuclidean flate geometry.
The more expansion that is produced when the symmetry breaks toward a
lower symmetry flatter manifold.
The Dedekind zeta function ( a kind of L function) is a special kind of
number theory function that relates the counting numbers to the
numbers produced by the mandifold irreducable polynomial in such a way
that it produces the vacuum load
or volume factor for the manifold.
The result is more closely related to volume integration over the
angular variables.
The Weeks manifold and the Thurston manifold are both knot-link related
manifolds
while E_8 ( Weeks relates this to thev Seifert-Weber space , I think)
and E_11
are related closed to the dodecahedron and higher Platonic type group
symmetries called
E=m*c^2/(4*Pi*r^3/3)
Eh=m*c^2/((4*Pi*r3/3)*( 1+k*R02/r2))
The volume is smaller the energy density is larger.
So the larger the R0 factor the smaller the volume and the bigger the
big bang explosion that results.
So calculating the volume right is important if you want to compare it
to observations
of the expansion of the universe ( Hubble's constant for instance).

[Knecht]

Post by Roger Bagula
The Zeta function I used was wrong,
 but no good Dedekind zeta for a Caratn E_11 polynomial based number
field actually exists in Mathematica as far as I can find:http://www.google.com/codesearch?hl=en&q=Dedekind+zeta+lang:mathemati...
Gives a cyclotomic number field version....
Hurwitz generralized version of Zeta is slightly smaller volume for
N[3*23^(3/2)Zeta[2, -x /. NSolve[x^3 - x + 1 == 0,
x][[1]]]/(4*Pi^4)]=0.938754
N[3*23^(3/2)Zeta[2, x /. NSolve[f[x] == 0, x][[11]]]/(4*Pi^4)]=0.240318
Which is way too low!
V=(4*Pi*r3/3)*( 1+k*R02/r2)
k=-1 for hyperbolic negative curvature
k=0 Euclidean zero curvature
k=1 for sphere elliptic plus curvature.,
R0 is the Manifold  reference curvature number.
ds^2=c^2-R(t)^2*(dr^2/(1-k*r^2)+r^2*dtheta^2+r^2*Sin[theta]^2*dPhi^2)
(In Einstein  and Riemann you see a factor of r^2/4).
In other words in an elliptical space the volume is more than a flate
Euclidean
and in a hyperbolic it is less.
The R0 factor is manifold or symmetry group dependant.
The Weeks space has the highest known R0 for a hyperbolic space.
The higher the R0 number the lower the volume
and the more vacuum energy the space has over a WEuclidean flate geometry.
The more expansion that is produced when the symmetry breaks toward a
lower symmetry flatter manifold.
The Dedekind zeta function  ( a kind of L function) is a special kind of
number theory function that relates the counting numbers to the
numbers produced by the mandifold irreducable polynomial in such a way
that it produces the vacuum load
or volume factor for the manifold.
The result is more closely related to volume integration over the
angular variables.
The Weeks manifold and the Thurston manifold are both knot-link related
manifolds
while E_8  ( Weeks relates this to thev Seifert-Weber space , I think)
and E_11
are related closed to the dodecahedron and higher Platonic type group
symmetries called
E=m*c^2/(4*Pi*r^3/3)
Eh=m*c^2/((4*Pi*r3/3)*( 1+k*R02/r2))
The volume is smaller the energy density is larger.
So the larger the R0 factor the smaller the volume and the bigger the
big bang explosion that results.
So calculating the volume right is important if you want to compare it
to observations
of the expansion of the universe ( Hubble's constant for instance).
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Can the great Oz tell us what the dark matter is, or why galaxies
exist at all, or why the observable universe is expanding, or why
galactic clusters seem to be anisotropically heading in one direction
with a scale length possibly exceeding the horizon length, or why
magnetic monopoles were never discovered, or what gamma-ray bursts are
all about, ...? I mean it is dandy to drop a lot of technical names
and terms, but behind the dubious and possibly demented analytical
staging is there a sensible, coherent cosmological paradigm that
explains what we observe? Or just a poser with a collection of
Ptolemaic epicycles?

Just curious,
Knecht
www.amherst.edu/~rloldershaw

[Knecht]

Post by Knecht
Can the great Oz tell us what the dark matter is, or why galaxies
exist at all, or why the observable universe is expanding, or why
galactic clusters seem to be anisotropically heading in one direction
with a scale length possibly exceeding the horizon length, or why
magnetic monopoles were never discovered, or what gamma-ray bursts are
all about, ...? I mean it is dandy to drop a lot of technical names
and terms, but behind the dubious and possibly demented analytical
staging is there a sensible, coherent cosmological paradigm that
explains what we observe? Or just a poser with a collection of
Ptolemaic epicycles?
Just curious,
Knechtwww.amherst.edu/~rloldershaw

I forgot to include a couple of things in the last post.

Can the Great Oz also please explain how the Substandard Paradigms of
particle physics and cosmology can possibly explain the "hierarchy
problem" which means among other things that the conventional Planck
mass and the estimated mass range for the "Higgs boson" differ
inexplicably and inappropriately by seventeen orders of magnitude!?

We would also like to see the Great Oz and Substandard Paradigm
proponents tell us more about the fact that the vacuum energy density
and the cosmological constant conflict to the tune of about 106 orders
of magnitude (and no that is not a misprint; the discrepancy is
usually cited as 10^106, give or take a few orders of magnitude. Well,
if "string theory" can postulate 10^500 different universes, what's
wrong with being of by a mere 10^106, right?

On the other hand, maybe the Substandard Paradigm has no clothes, so
to speak?

Yours in science,
Knecht
www.amherst.edu/~rloldershaw