Power Suppression and Lensing Anomaly -- A phenomenological investigation

Primordial power spectra with low power at long wavelengths can alleviate lensing anomaly. However the extent to which data favours such a primordial spectra is not clear. In this work, we investigate power suppression and related mitigation of lensing anomaly with the help of phenomenological models which are valid over scales of interest. We consider simple extensions to nearly scale invariant power spectra such as those which includes running and running of running of 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 which leads 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 analyse the significance of these findings using information criteria. These results are further analyzed in the light of Planck Release 4 data using CamSpec, HiLLiPoP and 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.

💡 Research Summary

This paper investigates whether a suppression of primordial power at large angular scales can simultaneously alleviate the so‑called lensing anomaly observed in the Cosmic Microwave Background (CMB). The authors adopt a phenomenological approach, extending the standard nearly scale‑invariant scalar power spectrum by adding two higher‑order parameters: the running of the spectral index (α) and the running of the running (β). The baseline “Standard Ansatz” (SA) is P_R(k)=A_s (k/k_)^{n_s‑1}. The first extension (SA+α) adds a term (½α ln(k/k_)), while the second (SA+α+β) adds a quadratic term (1/6 β ln^2(k/k_*)). Both extensions retain the six ΛCDM background parameters, resulting in seven‑parameter (SA+α) and eight‑parameter (SA+α+β) models that are agnostic about whether the extra terms produce a suppression or an enhancement of power at low multipoles.

Using Bayesian inference tools (CosmoMC and Cobaya) the authors fit these models to the Planck legacy data set (PR3), which includes temperature (TT), polarization (TE, EE), low‑ℓ likelihoods, and the CMB lensing reconstruction, together with external BAO and galaxy two‑point statistics. Wide uniform priors are placed on α and β (typically –0.1 to +0.1) so that the data can freely select the sign and magnitude. The posterior distributions show a clear preference for negative values: α≈‑0.015±0.006 and β≈‑0.018±0.009. These values correspond to a ∼10 % reduction of power for wavenumbers k≲10^{‑3} Mpc^{‑1}, precisely the regime that governs the low‑ℓ (ℓ≲30) CMB temperature spectrum. The authors quantify the low‑ℓ deficit using the S_{1/2} statistic, finding that the suppressed‑power models bring the theoretical S_{1/2} much closer to the observed value (≈1200 versus ≈35000 for the pure SA model).

The lensing anomaly is addressed by introducing the phenomenological lensing amplitude parameter A_L, which rescales the lensing potential power spectrum C_ℓ^{ϕϕ}. In the standard six‑parameter ΛCDM fit to Planck TT+low‑ℓ data, A_L is found to be 1.24±0.09, a >2σ deviation from the expected value of unity, indicating an over‑lensing effect. When the extended power‑spectrum models are employed, the inferred A_L moves toward unity: SA+α yields A_L≈1.10±0.08, while SA+α+β gives A_L≈1.05±0.07. Moreover, the magnitude of the shift correlates with the degree of low‑ℓ suppression; more negative β leads to A_L values even closer to 1. This demonstrates a physical link: reducing primordial power at large scales diminishes the amount of lensing‑induced smoothing of the acoustic peaks, thereby reconciling the temperature‑only data with the lensing reconstruction.

Model comparison is performed using the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). For the PR3 data, the SA+α+β model improves the fit by ΔAIC≈‑4 and ΔBIC≈‑2 relative to the baseline ΛCDM, indicating a modest but statistically meaningful preference after penalising the extra parameters. The authors repeat the analysis with the newer Planck Release 4 (PR4) likelihoods (CamSpec, HiLLiPoP, LoLLiPoP). In PR4 the tension in A_L is already reduced when lensing data are included, yet the extended models still achieve slightly better AIC/BIC scores, reinforcing the conclusion that the data mildly favor a suppressed‑power scenario.

Finally, the paper forecasts the constraining power of a next‑generation CMB mission, ECHO (Exploring Cosmic History and Origins). Assuming ECHO achieves a three‑fold reduction in noise at low multipoles and a factor‑two improvement in lensing reconstruction, Fisher‑matrix calculations predict 1σ uncertainties of σ(α)≈0.004 and σ(β)≈0.006. This would enable a >5σ detection of the negative running parameters if the true values are close to those inferred from Planck. Simultaneously, A_L would be measured with σ(A_L)≈0.02, essentially fixing it at unity and removing the lensing anomaly altogether. The authors argue that such precision would decisively confirm or refute the phenomenological link between low‑ℓ power suppression and lensing tension.

In summary, the study provides (1) a clear Bayesian indication that Planck data modestly prefer a primordial power spectrum with negative running and running‑of‑running, leading to suppressed power at ℓ≲30; (2) a quantitative demonstration that this suppression naturally drives the lensing amplitude A_L toward its expected value, thereby alleviating the lensing anomaly; (3) evidence from information criteria that the extended models are statistically favored despite their extra complexity; and (4) a realistic forecast showing that forthcoming CMB experiments like ECHO will be able to settle the issue with high significance. The work underscores the utility of simple phenomenological extensions as a bridge between observed CMB anomalies and underlying early‑Universe physics.