Planck intermediate results. LI. Features in the cosmic microwave background temperature power spectrum and shifts in cosmological parameters

TL;DR

The paper investigates why Planck TT data induce shifts in the ΛCDM parameters when comparing low-ℓ information (∼ℓ<800) to the full-ℓ spectrum (∼ℓ<2500). Using Planck TT with a τ prior and a suite of simulations, the authors quantify the significance of these shifts and develop a physical narrative connecting features in the high-ℓ power spectrum to changes in ω_m, ω_b, θ_*, n_s, and A_s e^{-2τ}, including the roles of lensing and the low-ℓ deficit. They find the observed shifts are consistent with statistical expectations once covariances and look-elsewhere effects are accounted for, and they show that most shifts arise from non-lensing physics rather than excessive peak smoothing. The study’s robustness tests and cross-checks with polarization, SPT, and Planck lensing support internal consistency of the Planck data within ΛCDM, reducing the need for new physics to explain the shifts and clarifying the data-driven origins of parameter inference differences.

Abstract

The six parameters of the standard $Λ$CDM model have best-fit values derived from the Planck temperature power spectrum that are shifted somewhat from the best-fit values derived from WMAP data. These shifts are driven by features in the Planck temperature power spectrum at angular scales that had never before been measured to cosmic-variance level precision. We investigate these shifts to determine whether they are within the range of expectation and to understand their origin in the data. Taking our parameter set to be the optical depth of the reionized intergalactic medium $τ$, the baryon density $ω_{\rm b}$, the matter density $ω_{\rm m}$, the angular size of the sound horizon $θ_*$, the spectral index of the primordial power spectrum, $n_{\rm s}$, and $A_{\rm s}e^{-2τ}$ (where $A_{\rm s}$ is the amplitude of the primordial power spectrum), we examine the change in best-fit values between a WMAP-like large angular-scale data set (with multipole moment $\ell<800$ in the Planck temperature power spectrum) and an all angular-scale data set ($\ell<2500$ Planck temperature power spectrum), each with a prior on $τ$ of $0.07\pm0.02$. We find that the shifts, in units of the 1$σ$ expected dispersion for each parameter, are $\{Δτ, ΔA_{\rm s} e^{-2τ}, Δn_{\rm s}, Δω_{\rm m}, Δω_{\rm b}, Δθ_*\} = \{-1.7, -2.2, 1.2, -2.0, 1.1, 0.9\}$, with a $χ^2$ value of 8.0. We find that this $χ^2$ value is exceeded in 15% of our simulated data sets, and that a parameter deviates by more than 2.2$σ$ in 9% of simulated data sets, meaning that the shifts are not unusually large. Comparing $\ell<800$ instead to $\ell>800$, or splitting at a different multipole, yields similar results. We examine the $\ell<800$ model residuals in the $\ell>800$ power spectrum data and find that the features there... [abridged]

Figures (15)

• Figure 1: Cosmological parameter constraints from PlanckTT+$\tau$prior for the full multipole range (orange) and for $\ell{<}800$ (blue)---see the text for the definitions of the parameters. Note that the constraints are generally in good agreement, with the full Planck data providing tighter limits on the parameters; however, the best-fit values certainly do shift. It is these shifts that we seek to explain in this paper. A prior $\tau = 0.07 \pm 0.02$ has been used here as a proxy for the effect of the low-$\ell$ polarization data (with the impact of a different prior discussed later). As a comparison, we also show results for WMAP $TT$ data combined with the same prior on $\tau$ (grey).
• Figure 2:
• Figure 3: Visually it might seem that the data point in the 6-parameter space of Fig. \ref{['fig:sims_cloud']} is a much worse outlier than only 1.4$\sigma$. One way to see that it really is only 1.4$\sigma$ is to transform to another parameter space, as shown in this figure. Linear transformations leave the $\chi^2$ unaffected, and while ours here are not exactly linear, the shifts are small enough that they can be approximated as linear and the $\chi^2$ is largely unchanged (in fact it is slightly worse, 1.6$\sigma$). We have chosen these parameters so the shifts are more decorrelated while still using physical quantities. The parameter $\tilde{A}_{\rm s}$ is the amplitude at a pivot of scale of $k=0.035{\rm Mpc}^{-1}$, chosen since there is no shift in $\tilde{A}_{\rm s}e^{-2\tau}$. Tick marks are omitted here for clarity.
• Figure 4: Distribution of two different statistics computed on the simulations (blue histogram) and on the data (orange line). The first is the $\chi^2$ statistic, where we compute $\chi^2$ for the change in parameters between $\ell{<}800$ and $\ell{<}2500$, with respect to the covariance of the expected shifts. The second is a "biggest outlier" statistic, where we search for the parameter with the largest change, in units of the standard deviation of the simulated shifts. We give the probability to exceed (PTE) on each panel. For both statistics, we find that the observed shifts are largely consistent with expectations from simulations.
• Figure 5: Response of ${\cal D}_l^{TT}$ ($\equiv\ell(\ell+1)C_\ell/2\pi$) to 1$\sigma$ increases in each of the parameters Lewis1999. All changes are made with the other five parameters pictured here held fixed. The dashed orange line in each panel shows the contribution from gravitational lensing alone. Note that the $y$-axis scale changes in some of the panels at $\ell{=}800$.