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.
