| Summary: | We point out that a modified temperature–redshift relation (<i>T</i>-<i>z</i> 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 <i>T</i> of the excited states in a probe environment of independently determined redshift <i>z</i>. 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 <i>T</i>-<i>z</i> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mi>h</mi><mi>ν</mi></mrow><mrow><msub><mi>k</mi><mi>B</mi></msub><mi>T</mi></mrow></mfrac></mrow></semantics></math></inline-formula>. Since the literature on extractions of the CMB’s <i>T</i>-<i>z</i> relation always assumes (i) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>z</mi><mo>)</mo><mi>ν</mi><mo>(</mo><mi>z</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>(</mo><mi>z</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula> is the observed frequency in the heliocentric rest frame, the finding (ii) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>z</mi><mo>)</mo><mi>T</mi><mo>(</mo><mi>z</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula> just confirms the expected blackbody nature of the interacting CMB at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. In contrast to the emission of isolated, directed radiation, whose frequency–redshift relation (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula>-<i>z</i> relation) is subject to (i), a non-conventional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula>-<i>z</i> relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo><mi>ν</mi><mo>(</mo><mi>z</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula> of pure, isotropic blackbody radiation, subject to adiabatically slow cosmic expansion, necessarily has to follow that of the <i>T</i>-<i>z</i> relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo><mi>T</mi><mo>(</mo><mi>z</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula> and vice versa. In general, the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><msup><mfenced separators="" open="(" close=")"><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></mfenced><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>≫</mo><mn>1</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> is non-linear for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>∼</mo><mn>1</mn></mrow></semantics></math></inline-formula>.
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