Remarks on Dynamical Dark Energy Measured by the Conformal Age of the Universe

We elaborate on a model of conformal dark energy (dynamical dark energy measured by the conformal age of the universe) recently proposed in [H. Wei and R.G. Cai, arXiv:0708.0884] where the present-day dark energy density was taken to be $\rho_q \equiv 3 \alpha^2 m_P^2/\eta^2$, where $\eta$ is the conformal time and $\alpha$ is a numerical constant. In the absence of an interaction between the ordinary matter and dark energy field $q$, the model may be adjusted to the present values of the dark energy density fraction $\Omega\Z{q} \simeq 0.73$ and the equation of state parameter $w\Z{q} < -0.78$, if the numerical constant $\alpha$ takes a reasonably large value, $\alpha\gtrsim 2.6$. However, in the presence of a nontrivial gravitational coupling of $q$-field to matter, say $\widetilde{Q}$, the model may be adjusted to the values $\Omega\Z{q}\simeq 0.73$ and $w\Z{q}\simeq -1$, even if $\alpha\sim {\cal O}(1)$, given that the present value of $\widetilde{Q}$ is large. Unlike for the model in [R.G. Cai, arXiv:0707.4049], the bound $\Omega\Z{q} <0.1$ during big bang nucleosynthesis (BBN) may be satisfied for almost any value of $\alpha$. Here we discuss some other limitations of this proposal as a viable dark energy model. The model draws some parallels with the holographic dark energy; we also briefly comment on the latter model.