Deconstructing the Planck TT Power Spectrum to Constrain Deviations from $\Lambda$CDM

Consistency checks of $\Lambda$CDM predictions with current cosmological data sets may illuminate the types of changes needed to resolve cosmological tensions. To this end, we modify the CLASS Boltzmann code to create phenomenological amplitudes, similar to the lensing amplitude parameter $A_L$, for the Sachs-Wolfe, Doppler, early Integrated Sachs-Wolfe (eISW), and Polarization contributions to the CMB temperature anisotropy, and then we include these additional amplitudes in fits to the Planck TT power spectrum. We find that allowing one of these amplitudes to vary at a time results in little improvement over $\Lambda$CDM alone suggesting that each of these physical effects are being correctly accounted for given the current level of precision. Further, we find that the only pair of phenomenological amplitudes that results in a significant improvement to the fit to Planck temperature data results from varying the amplitudes of the Sachs-Wolfe and Doppler effects simultaneously. However, we show that this model is really just refinding the $\Lambda$CDM + $A_L$ solution. We test adding our phenomenological amplitudes as well as $N_{\textrm{eff}}$, $Y_{\textrm{He}}$, and $n_{\textrm{run}}$ to $\Lambda$CDM + $A_L$ and find that none of these model extensions provide significant improvement over $\Lambda$CDM + $A_L$ when fitting Planck temperature data. Finally, we quantify the contributions of both the eISW effect and lensing on the constraint of the physical matter density from Planck temperature data by allowing the phenomenological amplitude from each effect to vary. We find that these effects play a relatively small role (the uncertainty increases by $3.5\%$ and $16\%$ respectively) suggesting that the overall photon envelope has the greatest constraining power.