Science > Physics > Quantum Gravity Via Expansion-Contraction 68.0: U.K. and ASU Rescue Brane Theory and Quintessence Via PI
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Science > Physics |
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"OsherD" |
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07 Jan 2007 10:54:57 PM |
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Quantum Gravity Via Expansion-Contraction 68.0: U.K. and ASU Rescue Brane Theory and Quintessence Via PI |
From Osher Doctorow
Carl L. Gardner of Arizona State U. (ASU) USA, basing his paper
especially on two U.K. papers from U. Sussex and U. Portsmouth, has
shown that simulations based on the sum of two exponentials as well as
on cosh (hyperbolic cosine, which is also the sum of two exponentials
of special type (1/2)(exp(u) + exp(-u)), give a good description of the
potential for the entire history and "prehistory" of the Universe. See
Carl L. Gardner, "Braneworld quintessential inflation and sum of
exponentials potentials," hep-ph/0701036 v1 4 Jan 2007, 20 pages.
The U. K. papers are astro-ph/0006421 and a second paper in arXiv whose
citation number I forgot but whose title is "Chaotic inflation on
brane," by R. Maarten, D. Wands, and B. Bassett of U. Portsmouth U.K.
The first paper is by E. J. Copeland, A. R. Liddle, J. E. Lidsey and
was published in Phys. Rev. D 64, 023509 (2001).
The sum of two exponential functions is the sum of two solutions to the
Riccati Differential equation dy/dt = A(t) + B(t)y + C(t)y^2 (with
possibly different coefficients for two Riccati Differential equations,
so y1 and y2 would be a better designation), which are "Fundamental
Equations of Probable Influence/Causation (PI)" although the authors of
the above-cited papers were unfamiliar with PI presumably. Here B(t)
and A(t) = 0.
Although the sum of two exponential functions satisfying different
Riccati Differential equations is not itself ordinarily a solution of a
Riccati Differential equation because of the nonlinearity of the latter
in general even when only B(t) is a nonzero coefficient, we still have
PI dynamics in each of them, and we can always regard the result as a
"generalized Riccati Differential equation" of form:
1) d(y1 + y2)/dt = B1(t)y1 + B2(t)y2
Gardner's paper at the same time rescurrects Quintessence in the form
of Quintessential Inflation (a single scalar field playing the role of
the Inflaton and simultaneously of Quintessence).
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity Via Expansion-Contraction 68.0: U.K. and ASU Rescue Brane Theory and Quintessence Via PI |
07 Jan 2007 11:11:18 PM |
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From Osher Doctorow
Gardner's sum-of-exponentials potential is in 5-dimensional gravitation
with standard model particles confined to "our" 3-brane. He points out
that braneworld quintessential inflation can occur for potentials with
or without a minimum and with or without external acceleration and an
event horizon. With a modified time scale variable z, the transitions
to acceleration due to quintessence occurs near z = 1, and the low z
behavior of the equation of state parameter provides a clear observable
signal that distinguishes quintessence from a cosmological constant.
He points out that a sum of exponentials occurs naturally in M/String
theory for combinations of moduli fields or combinations of dilaton and
moduli fields. The potential V is a function of the scalar field phi,
V(phi), and for all 4 potentials (sums of exponentials and special sum
of exponentials which is cosh) considered, V is approximately
Aexp(lambda phi) for phi >> 1 (much greater than 1) which occurs during
inflation and gravitational particle production.
The homogeneous scalar field obeys the Klein-Gordon equation since it
is confined to the brane and a braneworld modified Friedmann equation
from the U.K. references with a constant embodying the effects of bulk
gravitons on brane and a 4-dimensional cosmological constant and a
4-dimensional brane tension. During inflation, the "dark radiation
term" involving the effects of bulk gravitons on the brane rapidly goes
to 0 so that term can be ignored.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity Via Expansion-Contraction 68.0: U.K. and ASU Rescue Brane Theory and Quintessence Via PI |
07 Jan 2007 11:22:23 PM |
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From Osher Doctorow
I meant to say in the first post of this Section that A(t) and C(t) are
0 but not B(t) in the Riccati Differential equation considered here.
The three main models that Gardner considers are:
1) V(phi) = A(exp(5 phi) + exp(sqrt(2) phi)
2) V = A(exp(5 phi) + exp(phi))
3) V = 2A cosh(5 phi)
He also considers a 4th model with just one exponential.
Osher Doctorow
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