Estimation of the CMB temperature from atomic C {\sc i} and molecular CO lines in the interstellar medium of early galaxies

The linear increase of the cosmic microwave background (CMB) temperature with cosmological redshift, $T_{\rm CMB} = T_0(1 + z)$, is a prediction of the standard cosmological $\Lambda$CDM model. There are currently two methods to measure this dependence at redshift $z>0$, and that is equally important to estimate the CMB temperature $T_0$ at the present epoch $z=0$. The first method is based on the Sunyaev-Zeldovich (SZ) effect for a galaxy cluster. aThe second method is based on the analysis of the populations of atomic and molecular energy levels observed in the absorption spectra of quasars. This method allows $T_{\rm CMB}(z)$ to be measured directly. We present new estimates of $T_{\rm CMB}(z_i)$ in the redshift range $1.7\le z_i \le3.3$ based on the analysis of excitation of the CO rotational levels and C {\sc i} fine-structure levels in 15 absorption systems. We take into account collisional excitation of CO and C {\sc i} with hydrogen atoms and H$_2$ and radiative pumping of C {\sc i} by the interstellar ultraviolet radiation. Applying this corrections leads to a systematic decrease in the previously obtained estimates of $T_{\rm CMB}(z_i)$ (for some systems the magnitude of the effect is $\sim$10%). Combining our measurements with the measurements of $T_{\rm CMB}(z)$ in galaxy clusters we have obtained a constraint on the parameter $\beta=+0.010\pm0.013$, which characterizes the deviation of the CMB temperature from the standard relation, $T_{\rm CMB} = T_0(1 + z)^{1-\beta}$, and an independent estimate of the CMB temperature at the present epoch, $T_0 = 2.719\pm0.009$ K, which agrees well with the estimate from orbital measurements, $T_0 = 2.7255\pm0.0006$ K. This independent estimate is very important because it was obtained using cosmological data, in contrast to satellite measurements, which are obtained "here" and "now".