The general property of dynamical quintessence field

We discuss the general dynamical behaviors of quintessence field, in particular, the general conditions for tracking and thawing solutions are discussed. We explain what the tracking solutions mean and in what sense the results depend on the initial conditions. Based on the definition of tracking solution, we give a simple explanation on the existence of a general relation between $w_\phi$ and $\Omega_\phi$ which is independent of the initial conditions for the tracking solution. A more general tracker theorem which requires large initial values of the roll parameter is then proposed. To get thawing solutions, the initial value of the roll parameter needs to be small. The power-law and pseudo-Nambu Goldstone boson potentials are used to discuss the tracking and thawing solutions. A more general $w_\phi-\Omega_\phi$ relation is derived for the thawing solutions. Based on the asymptotical behavior of the $w_\phi-\Omega_\phi$ relation, the flow parameter is used to give an upper limit on $w_\phi'$ for the thawing solutions. If we use the observational constraint $w_{\phi 0}<-0.8$ and $0.2<\Omega_{m0}<0.4$, then we require $n\lesssim 1$ for the inverse power-law potential $V(\phi)=V_0(\phi/m_{pl})^{-n}$ with tracking solutions and the initial value of the roll parameter $|\lambda_i|<1.3$ for the potentials with the thawing solutions.Comment: 11 figures, corrected some typos and presentation improved, PLB in pres

Friedmann Equations and Thermodynamics of Apparent Horizons

With the help of a masslike function which has dimension of energy and equals to the Misner-Sharp mass at the apparent horizon, we show that the first law of thermodynamics of the apparent horizon $dE=T_AdS_A$ can be derived from the Friedmann equation in various theories of gravity, including the Einstein, Lovelock, nonlinear, and scalar-tensor theories. This result strongly suggests that the relationship between the first law of thermodynamics of the apparent horizon and the Friedmann equation is not just a simple coincidence, but rather a more profound physical connection.Comment: no figures, V2: re-organized and re-writtend, main results and conclusion unchanged, Phys. Rev. Lett. in pres

The challenge for single field inflation with BICEP2 result

The detection of B-mode power spectrum by the BICEP2 collaboration constrains the tensor-to-scalar ratio $r=0.20^{+0.07}_{-0.05}$ for the lensed-$\Lambda$CDM model. The consistency of this big value with the {\em Planck} results requires a large running of the spectral index. The large values of the tensor-to-scalar ratio and the running of the spectral index put a challenge to single field inflation. For the chaotic inflation, the larger the value of the tensor-to-scalar ratio is, the smaller the value of the running of the spectral index is. For the natural inflation, the absolute value of the running of the spectral index has an upper limit.Comment: 4 figures, 3 pages, some references added, PLB in pres