By EDMUND J COPELAND; SAMI M; TSUJIKAWA SHINJI; D V Ahluwalia-Khalilova
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Extra info for DYNAMICS OF DARK ENERGY
In Ref. , the authors consider an axion model which leads to a time dependent equation of state parameter w(z) for the Quintessence field. As before, the small mass scale is protected against radiative corrections. The novel feature they introduce is the seesaw mechanism, which allows for two natural scales to play a vital role in determining all the other fundamental scales. These are the weak scale, v, and the Planck scale, Mpl . For ex1/4 ample, the dark energy density ρDE ∝ v 2 /Mpl , and the 3 radiatively stable mass mφ ∝ v 4 /Mpl .
First integrate out (exactly) all of the degrees of freedom on the branes, giving the low-energy brane dependence on the massless 4D graviton mode. The effect of this is to obtain a large effective brane tension, T ∼ Mg4 for each of the 3-branes which might be present, which includes the vacuum energies of all of the presently-observed elementary particles. Following this, there is a classical part of the integration over the bulk degrees of freedom. This is achieved by solving the classical supergravity equations to determine how the extra dimensions curve in response to the brane sources which are scattered throughout the extra dimensions.
Hence the phantom fixed point (b) is a stable √ spiral. The point (c) is a stable node √ for 0 < λ < 3, whereas it is√a saddle point for λ > 3. This critical value λ∗ = 3 is computed by setting the determinant of M to √ be zero. The point (d) is physically meaningful for λ > 3 because of the condition Ωφ < 1, and it is a stable node . Hence the point (d) is stable when the point (c) is unstable and vice versa. It was shown in Ref.  that this property holds for all scalar-field models which possess scaling solutions.
DYNAMICS OF DARK ENERGY by EDMUND J COPELAND; SAMI M; TSUJIKAWA SHINJI; D V Ahluwalia-Khalilova