  We point out that a modified temperature-redshift relation ($T$-$z$ relation)
of the cosmic microwave background (CMB) can not be deduced by any
observational method that appeals to an a priori thermalisation to the CMB
temperature $T$ of the excited states in a probe environment of independently
determined redshift $z$. For example, this applies to quasar-light absorption
by a damped Lyman-alpha system due to atomic as well as ionic fine-splitting
transitions or molecular rotational bands. Similarly, the thermal
Sunyaev-Zel'dovich (thSZ) effect cannot be used to extract the CMB's $T$-$z$
relation. This is because the relative line strengths between ground and
excited states in the former and the CMB spectral distortion in the latter case
both depend, apart from environment specific normalisations, solely on the
dimensionless spectral variable $x=\frac{h\nu}{k_B T}$. Since literature on
extractions of the CMB's $T$-$z$ relation always assumes (i)
$\nu(z)=(1+z)\nu(z=0)$ where $\nu(z=0)$ is the observed frequency in the
heliocentric rest frame, the finding (ii) $T(z)=(1+z)T(z=0)$ just confirms the
expected blackbody nature of the interacting CMB at $z>0$. In contrast to
emission of isolated, directed radiation, whose frequency-redshift relation
($\nu$-$z$ relation) is subject to (i), a non-conventional $\nu$-$z$ relation
$\nu(z)=f(z)\nu(z=0)$ of pure, isotropic blackbody radiation, subject to
adiabatically slow cosmic expansion, necessarily has to follow that of the
$T$-$z$ relation $T(z)=f(z)T(z=0)$ and vice versa. In general, the function
$f(z)$ is determined by energy conservation of the CMB fluid in a
Friedmann-Lemaitre-Robertson-Walker universe. If the pure CMB is subject to an
SU(2) rather than a U(1) gauge principle, then $f(z)=
\left({1/4}\right)^{1/3}(1+z)$ for $z\gg 1$, and $f(z)$ is nonlinear for $z\sim
1$.
