Primordial power spectra with low power at long wavelengths can alleviate lensing anomaly. However the extent to which data favors such a primordial spectrum is not clear. In this work, we investigate power suppression and related mitigation of lensing anomaly with the help of phenomenological models that are valid over scales of interest. We consider simple extensions to nearly scale invariant power spectra such as those that include running and running of running of the spectral index. We perform Bayesian analysis of these models, which are agnostic about power suppression, with Planck legacy data and show that data tend to choose parameters that lead to power suppression at low multipoles. We then investigate the connection between power suppression and alleviation of lensing anomaly and show that lensing anomaly is mitigated the most in models with maximum suppression of power at low multipoles. We also analyze the significance of these findings using information criteria. These results are further analyzed in the light of Planck Release 4 data using CamSpec, High-$\ell$ Likelihood on Polarized Power spectra (HiLLiPoP), and Low-$\ell$ Likelihood on Polarized Power spectra (LoLLiPoP) likelihoods in which departure of lensing parameter from one is significantly reduced. Furthermore, we investigate the ability of near-ultimate future CMB missions such as ECHO to put tighter constraints on these models and to settle the issue. We conclude that we can make stronger conclusions about the presence of power suppression in the future by studying such simple phenomenological models.

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• Cosmic microwave background
• Cosmological parameters
• Inflation