We point out that a modified temperature–redshift relation (
T-
z relation) of the cosmic microwave background (CMB) cannot 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
. Since the literature on extractions of the CMB’s
T-
z relation always assumes (i)
, where
is the observed frequency in the heliocentric rest frame, the finding (ii)
just confirms the expected blackbody nature of the interacting CMB at
. In contrast to the emission of isolated, directed radiation, whose frequency–redshift relation (
-
z relation) is subject to (i), a non-conventional
-
z relation
of pure, isotropic blackbody radiation, subject to adiabatically slow cosmic expansion, necessarily has to follow that of the
T-
z relation
and vice versa. In general, the function
is determined by the 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
for
, and
is non-linear for
.
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