1. Dark Energy and the Inflection Points of Cosmic Expansion in Standard and Brane Cosmologies
Daniel Schmidt, Liberty University
Cyclotron Institute —Texas A&M University
Mentor: Dr. Akram Zhanov

2. The Friedman Equations (Standard Cosmology)
Here R(t) is the scale factor, representing the “size” of the universe, Λ is the cosmological constant, or vacuum energy density, k is the curvature of space, p is pressure, and ρ is energy density.
The second equation is separable, (since k=0, according to observations) so we can solve for t as a function of Λ.

3. The Importance of Acceleration—The Age of the Universe
Upper and lower limits on t:
tmin = 12 Ga λ = 0.59
tmax = 17 Ga λ = 0.89
Currently accepted value:
tWMAP = 13.7 Ga λ = 0.75
Note: λ = Λ 8 π G ρc is a dimensionless “reduced” cosmological constant, where ρc is the critical density. (1 Ga = 1 billion years)

4. Observational Evidence for Nonzero λ
Without λ, the universe would be younger than some of its oldest stars.
Other evidence for λ 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 λ on the expansion.

5. Measured Values of λ Paper Technique Value Limit Kochanek 1996
Gravitational lensing statistics NA
<0.66 (95% confidence)
Myungshin et al. 1997
0.64 (+0.15, -0.26)
Chiba & Yoshii 1997
0.8
Chiba & Yoshii 1999
0.7 (+0.1, -0.2)
Perlmutter et al. 1997
High-Z supernovae
0.06 (+0.28, -0.34)
<0.51 (95% confidence)
Riess et al. 1998
0.68 (+-0.10)
0.84 (+-0.09)
>0 (98% confidence)
Perlmutter et al. 1999
0.71 (+0.08, -0.09)
>0 (99% confidence)

6. Measured Values of λ
Here the measured values of λ are superimposed on the t vs. λ graph.

7. 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 λ, 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.
This can be found by setting R’’ = 0, and solving for R.

8. Inflection Point in Standard Cosmology
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.)

9. Inflection Point(s) in Brane Cosmology
Superstring 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.
Here ρb is a constant introduced by the model.

10. Inflection Point(s) in Brane Cosmology
Multiply this by R8, differentiate, and set R’’ = 0.
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.

11. Inflection Point(s) in Brane Cosmology
The precise location of this inflection point cannot be determined, since the parameter ρb is not known. It depends on the five dimensional Plank mass as follows:
ρb = 96 π G M56
M5 is not known, but it may be constrained by this model.
The single inflection point occurs for x < 1 only if
M5 > 1.9 x 1018 eV/c2.
Thus, if the RSII model is correct, we have a constraint on M5

12. A Future Inflection Point?
In some models, dark energy has a non-constant energy density. A decreasing value of Λ could allow another inflection point in the future, when the cosmic repulsion eventually becomes too weak to overcome the gravitation of “ordinary” matter.
One such model proposed by Andrei Linde even allows Λ to become negative in the distant future, meaning that the expansion would eventually stop, and the universe would recollapse.
Similar ideas have revived the idea of a cyclic universe, in which a future “big crunch” is reversed and the universe begins to expand again, driven by a changing dark energy density.

13. This research was supported by NSF grant PHY , the Texas A&M Cyclotron Institute, and the Department of Energy.
Special thanks to Dr. Akram Zhanov and the Cyclotron Institute staff.