Cosmological constraints on the new holographic dark energy model with
  action principle
  Recently, a New HDE model with action principle was proposed (Li and Miao,
arXiv:1210.0966). This model completely solves the causality and circular
problems in the original HDE model, and is similar to the original model except
a new term that can be interpreted as dark radiation. In this paper, we make
further investigations on this model from the aspect of cosmological
observations. Numerically, we confirm that the equations of motion force the
$L(z=-1)=0$, making the cut-off $aL$ exactly the future event horizon. We also
perform detailed analysis on the dynamical properties of the model, divided
into the $c<6$ and $c\geq6$ cases ($c$ is a dimensionless parameter which
should be decided by the data). From a combination of the present
Union2.1+BAO+CMB+$H_0$ data, we find the model yields $\chi^2_{\rm
min}=548.798$ (in a non-flat Universe), comparable to the results of the
original HDE model (549.461) and the concordant $\Lambda$CDM model (550.354).
At 95.4% CL, we get $1.41<c<3.09$ and correspondingly $-2.25<w(z=-1)<-1.39$,
implying the Big Rip fate of the Universe at a high confidence level. Besides,
for the constraints on dark radiation, we also get a rough estimation $N_{\rm
\rm eff}=3.54^{+0.32+0.67}_{\rm -0.45-0.76}$, with the central value slightly
larger than the standard value 3.046.
