A theoretical framework is developed for treating the quantization of the photons in a spacetime with a longitudinal expansion. This can be used to study the production of the photons through the non-equilibrium relaxation of a disoriented chiral con
densate presumably formed in the expanding hot central region in ultra-relativistic heavy-ion collisions. These photons can be a signature of the formation of disoriented chiral condensates in the direct photon measurements of heavy-ion collisions.
We consider that the pre-inflation era is radiation-dominated, transiting smoothly to the inflationary era. We work out in detail the dynamics of inflaton fluctuations across the phase transition and the proper choices of initial vacuum states. It is
found that this phase transition can suppress long-wavelength quantum fluctuations of inflaton. This may attribute to the large-scale CMB anisotropy a lower power than predicted in the standard $Lambda$CDM model. In constraining this transitional effect by WMAP anisotropy data, we use the WMAP best-fit scale-invariant $Lambda$CDM model with the density power spectrum replaced by the one found in this work. We find that the transition occurs at least about 10 e-folds before the comoving scales comparable to our present horizon size cross the Hubble radius during inflation.
We discuss the ratio of the angular diameter distances from the source to the lens, $D_{ds}$, and to the observer at present, $D_{s}$, for various dark energy models. It is well known that the difference of $D_s$s between the models is apparent and t
his quantity is used for the analysis of Type Ia supernovae. However we investigate the difference between the ratio of the angular diameter distances for a cosmological constant, $(D_{ds}/D_{s})^{Lambda}$ and that for other dark energy models, $(D_{ds}/D_{s})^{rm{other}}$ in this paper. It has been known that there is lens model degeneracy in using strong gravitational lensing. Thus, we investigate the model independent observable quantity, Einstein radius ($theta_E$), which is proportional to both $D_{ds}/D_s$ and velocity dispersion squared, $sigma_v^2$. $D_{ds}/D_s$ values depend on the parameters of each dark energy model individually. However, $(D_{ds}/D_s)^{Lambda} - (D_{ds}/D_{s})^{rm{other}}$ for the various dark energy models, is well within the error of $sigma_v$ for most of the parameter spaces of the dark energy models. Thus, a single strong gravitational lensing by use of the Einstein radius may not be a proper method to investigate the property of dark energy. However, better understanding to the mass profile of clusters in the future or other methods related to arc statistics rather than the distances may be used for constraints on dark energy.
