We present an approximate, analytical calculation of the reionized spectra $C_l^{XX}$ of cosmic microwave background radiation (CMB) anisotropies and polarizations generated by relic gravitational waves (RGWs). Three simple models of reionization are
explored, whose visibility functions are fitted by gaussian type of functions as approximations. We have derived the analytical polarization $beta_l$ and temperature anisotropies $alpha_l$, both consisting of two terms proportional to RGWs at the decoupling and at the reionization as well. The explicit dependence of $beta_l$ and $alpha_l$ upon the reionization time $eta_r$, the duration $Deltaeta_r$, and the optical depth $kappa_r$ are demonstrated. Moreover, $beta_l$ and $alpha_l$ contain $kappa_r$ in different coefficients, and the polarization spectra $C_l^{EE}$ are $C_l^{BB}$ are more sensitive probes of reionization than $C_l^{TT}$. These results facilitate examination of the reionization effects, in particular, the degeneracies of $kappa_r$ with the normalization amplitude and with the initial spectral index of RGWs. It is also found that reionization also causes a $kappa_r$-dependent shift $Delta lsim 20$ of the zero multipole $l_0$ of $C_l^{TE}$, an effect that should be included in order to detect the traces of RGWs. Compared with numerical results, the analytical $C_l^{XX}$ as approximation have the limitation. For the primary peaks in the range $lsimeq (30, 600)$, the error is $le 3%$ in three models. In the range $l < 20$ for the reionization bumps, the error is $le 15%$ for $C_l^{EE}$ and $C_l^{BB}$ in the two extended reionization models, and $C_l^{TT}$ and $C_l^{TE}$ have much larger departures for $l<10$. The bumps in the sudden reionization model are too low.
We present an analytical calculation of the spectra of CMB anisotropies and polarizations generated by relic gravitational waves (RGWs). As a substantial extension to the previous studies, three new ingredients are included in this work. Firstly, the
analytic $C_l^{TT}$ and $C_l^{TE}$ are given; especially the latter can be useful to extract signal of RGWs from the observed data in the zero multipole method. Secondly, a fitting formula of the decaying factor on small scales is given, coming from the visibility function around the photon decoupling. Thirdly, the impacts by the neutrino free-streaming (NFS) is examined, a process that occurred in the early universe and leaves observable imprints on CMB via RGWs. It is found that the analytic $C_l^{TT}$ and $C_l^{TE}$ have profiles agreeing with the numeric ones, except that $C^{TT}_l$ in a range $l le 10$ and the $1^{st}$ trough of $C_l^{TE}$ around $l sim 75$ have some deviations. With the new damping factor, the analytic $C^{EE}_l$ and $C^{BB}_l$ match with the numeric ones with the maximum errors only $sim 3%$ up to the first three peaks for $lle 600$, improving the previous studies substantially. The correspondence of the positions of peaks of $C^{XX}_l$ and those of RGWs are also demonstrated explicitly. We also find that NFS reduces the amplitudes of $C^{XX}_l$ by $(20% sim 35%)$ for $lsimeq(100sim 600)$ and shifts slightly their peaks to smaller angles. Detailed analyses show that the zero multipoles $l_0$, where $C_l^{TE}$ crosses 0, are shifted to larger values by NFS. This shifting effect is as important as those causedby different inflation models and different baryon fractions.
In seeking a model solving the coincidence problem, the effective Yang-Mills condensate (YMC) is an alternative candidate for dark energy. A study is made for the model up to the 2-loop order of quantum corrections. It is found that, like in the 1-lo
op model, for generic initial conditions during the radiation era, there is always a desired tracking solution, yielding the current status $Omega_Lambda simeq 0.73$ and $Omega_m simeq 0.27$. As the time $tto infty$ the dynamics is a stable attractor. Thus the model naturally solves the coincidence problem of dark energy. Moreover, if YMC decays into matter, its equation of state (EoS) crosses -1 and takes $wsim -1.1$, as indicated by the recent observations.
