Kinetic-theory derivation and Bayesian analysis of the finite-temperature shifted CMB power spectrum

Finite-temperature quantum-gravitational corrections to the cosmological constant introduce two additional density parameters, ${\mathrm{\Omega }}_{{\mathrm{\Lambda }}_2}$ and ${\mathrm{\Omega }}_{{\mathrm{\Lambda }}_3}$, the associated terms of which scale as $a^{-4}$ and $a^{-2}$, respectively. In previous work (Papers I and II), these parameters were studied using machine-learning methods that identified a best-fit point. Here we complete the picture with two complementary analyses. First, we perform a Bayesian MCMC analysis using a modified version of $\text{textsc}class$ against the Planck 2018 likelihoods. This tells us what the data can constrain once all parameter uncertainties have been marginalized over. Both parameters are consistent with zero, with ${\mathrm{\Omega }}_{{\mathrm{\Lambda }}_3}$ constrained only from one side. Second, we build a CMB pipeline from scratch, following Weinberg’s kinetic-theory formulation and validating it against $\text{textsc}class$. This allows us to study the underlying physics. The response analysis carried out with it shows that ${\mathrm{\Omega }}_{{\mathrm{\Lambda }}_3}$ is almost entirely absorbed by a shift in $h$ over the well-measured acoustic multipoles. The part of the signal that cannot be absorbed resides at low $\mathrm{\ell }$, where cosmic variance dominates. Meanwhile, the ${\mathrm{\Omega }}_{{\mathrm{\Lambda }}_2}$ signature is nearly degenerate with the overall amplitude of the spectrum. The effects of both parameters on the CMB power spectrum are small, since they represent higher-order thermal corrections to the cosmological constant. The resulting constraints are correspondingly weak: the induced shifts are largely absorbed by degeneracies with $h$ and with the overall normalization of the spectrum. This is the reason why neither parameter is detected.