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The linear increase of the cosmic microwave background (CMB) temperature withcosmological redshift, $T_{\rm CMB} = T_0(1 + z)$, is a prediction of thestandard cosmological $\Lambda$CDM model. There are currently two methods tomeasure this dependence at redshift $z>0$, and that is equally important toestimate the CMB temperature $T_0$ at the present epoch $z=0$. The first methodis based on the Sunyaev-Zeldovich (SZ) effect for a galaxy cluster. aThe secondmethod is based on the analysis of the populations of atomic and molecularenergy levels observed in the absorption spectra of quasars. This method allows$T_{\rm CMB}(z)$ to be measured directly. We present new estimates of $T_{\rmCMB}(z_i)$ in the redshift range $1.7\le z_i \le3.3$ based on the analysis ofexcitation of the CO rotational levels and C\,{\sc i} fine-structure levels in15 absorption systems. We take into account collisional excitation of CO andC\,{\sc i} with hydrogen atoms and H$_2$ and radiative pumping of C\,{\sc i} bythe interstellar ultraviolet radiation. Applying this corrections leads to asystematic decrease in the previously obtained estimates of $T_{\rm CMB}(z_i)$(for some systems the magnitude of the effect is $\sim$10\%). Combining ourmeasurements with the measurements of $T_{\rm CMB}(z)$ in galaxy clusters wehave obtained a constraint on the parameter $\beta=+0.010\pm0.013$, whichcharacterizes 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 CMBtemperature at the present epoch, $T_0 = 2.719\pm0.009$\,K, which agrees wellwith the estimate from orbital measurements, $T_0 = 2.7255\pm0.0006$\,K. Thisindependent estimate is very important because it was obtained usingcosmological data, in contrast to satellite measurements, which are obtained"here" and "now".

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