Dark Energy and the Inflection Points of Cosmic Expansion in Standard and Brane Cosmologies

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Dark Energy and the Inflection Points of Cosmic
Expansion in Standard and Brane
Cosmologies Daniel Schmidt, Liberty
University Cyclotron Institute--Texas AM
University Mentor Dr. Akram Zhanov
Friedman Equations in Standard Cosmology
Observational Evidence for Nonzero ? Without
?, the universe would be younger than some of its
oldest stars. Other evidence for ? comes from
observations of high redshift supernovae, and
gravitational lensing statistics. Type Ia
supernovae are used as standard candles to
measure distance independently of redshift. This
allows the redshift-distance relation to be
measured, which in turn yields the history of
cosmic expansion. Gravitational lensing of high
redshift objects is another test. The number of
condensed, lensing bodies per unit volume depends
on the scaling factor. Measuring the evolution
of this number allows one to observe the effects
of ? on the expansion. Measured values are
summarized at right.
Upper and lower limits on t tmin 12 Ga ?
0.59 tmax 17 Ga ? 0.89 Currently accepted
value (from WMAP data) tWMAP 13.7 Ga ? 0.75
Paper Technique Value Limit
Kochanek 1996 Gravitational lensing statistics NA lt0.66 (95 confidence)
Myungshin et al. 1997 Gravitational lensing statistics 0.64 (0.15, -0.26) NA
Chiba Yoshii 1997 Gravitational lensing statistics 0.8 NA
Chiba Yoshii 1999 Gravitational lensing statistics 0.7 (0.1, -0.2) NA
Perlmutter et al. 1997 High-Z supernovae 0.06 (0.28, -0.34) lt0.51 (95 confidence)
Riess et al. 1998 High-Z supernovae 0.68 (-0.10) 0.84 (-0.09) gt0 (98 confidence)
Perlmutter et al. 1999 High-Z supernovae 0.71 (0.08, -0.09) gt0 (99 confidence)
These equations describe the evolution of the
scale factor R(t), which represents the relative
size of the universe. Here ? is the
cosmological constant, or vacuum energy density,
k is the curvature of space, p is pressure, and ?
is energy density. The second equation is
separable, so we can solve for t as a function of
?.
t vs. ? graph Note ? ? 8 p G ?c is a
dimensionless reduced cosmological constant,
where ?c is the critical density. (1 Ga 1
billion years)
Inflection Point in Standard Cosmology
The History of Cosmic Expansion Returning to
the first Friedman equation, we see that at
earlier times in the history of the universe,
the matter and radiation density would have been
more important than ?, leading to deceleration.
In the present universe, the expansion is known
to be accelerating. Somewhere in the middle,
there must exist an inflection point at which the
acceleration is momentarily zero. (See graph at
right) This can be found by setting R 0, and
solving for R. Note k has been set to zero
since observations indicate that the universe is
flat.
R/R0 vs. t
This quartic equation has only one positive real
solution R 0.60 R0 corresponding to a time of
about 7.2 Ga. (R0 is the current value of the
scale factor.)
Here the measured values of ? are superimposed on
the t vs. ? graph.
Inflection Point(s) in Brane Cosmology Superstri
ng theories introduce some corrections to the
Friedman equations, leading to possible changes
in the evolution of the scale factor R. The
second equation below is the corrected Freidman
equation according to the RSII model.
Multiply this by R8, differentiate, and set R
0.
The precise location of this inflection point
cannot be determined, since the parameter ?b is
not known. It depends on the five dimensional
Plank mass as follows ?b 96 p G M56 M5 is
not known, but it may be constrained by this
model. The single inflection point occurs for
x lt 1 only if M5 gt 1.9 x 1018 eV/c2. Thus, if
the RSII model is correct, we have a constraint
on M5
This research was supported by NSF grant
PHY463291-00001, the Texas AM Cyclotron
Institute, and the Department of Energy.
Special thanks to Dr. Akram Zhanov and the
Cyclotron Institute staff.
Here x R/R0. The resulting seventh order
polynomial equation has one real solution, as in
the standard cosmology. Thus, it seems that the
results found in the standard cosmology are valid
in brane cosmology as well.
Here ?b is a constant introduced by the model.