Cosmic Microwave Background Acoustic Peak Locations

TL;DR

This work develops an analytic framework to understand the acoustic peak locations in the CMB TT, EE, and TE power spectra within ΛCDM. Starting from a simple baseline model with tight coupling and a one-to-one k-to-ell projection, the authors quantify how various physical effects shift peak positions through phase shifts in photon perturbations and through the projection process. They decompose the total phase into decoupling, gravitational driving (both gamma and neutrino components), and then incorporate corrections from the LSS width, primordial power spectrum, projection asymmetry, ISW/dipole/polarization, and lensing. The resulting predictions are compared to Planck 2015 measurements, showing good agreement and offering physical intuition, with potential for fast analytic emulation of Boltzmann codes. This framework clarifies how complex microphysical processes imprint measurable shifts on the acoustic peaks, strengthening cosmological inferences from CMB data.

Abstract

The Planck collaboration has measured the temperature and polarization of the cosmic microwave background well enough to determine the locations of eight peaks in the temperature (TT) power spectrum, five peaks in the polarization (EE) power spectrum and twelve extrema in the cross (TE) power spectrum. The relative locations of these extrema give a striking, and beautiful, demonstration of what we expect from acoustic oscillations in the plasma; e.g., that EE peaks fall half way between TT peaks. We expect this because the temperature map is predominantly sourced by temperature variations in the last scattering surface, while the polarization map is predominantly sourced by gradients in the velocity field, and the harmonic oscillations have temperature and velocity 90 degrees out of phase. However, there are large differences in expectations for extrema locations from simple analytic models vs. numerical calculations. Here we quantitatively explore the origin of these differences in gravitational potential transients, neutrino free-streaming, the breakdown of tight coupling, the shape of the primordial power spectrum, details of the geometric projection from three to two dimensions, and the thickness of the last scattering surface. We also compare the peak locations determined from Planck measurements to expectations under the $Λ$CDM model. Taking into account how the peak locations were determined, we find them to be in agreement.

Figures (7)

• Figure 1: Comparison of the spectra of the fiducial cosmology (solid curves) and the peak locations predicted by the baseline model (vertical dashed lines).
• Figure 2: The intervals $\Delta(kr_s)$ of neighboring peak-trough of $\left[\Theta_0+\Phi\right](k, \eta)$ for mode $k=0.5 \ {\rm Mpc}^{-1}$. Dots are numerical results of peak-trough intervals. Solid line is the analytic result of high-order correction to the tight coupling approximation $\phi_{\rm dcp}$. Dashed line is the result of corrections from both late-time high-order correction $\phi_{\rm dcp}$ and early-time gravitational driving $\phi_{\rm gr,\gamma}$ sourced by photon perturbations.
• Figure 3: Phase shifts of sources induced by $3.046$ neutrinos and measured at the LSS.
• Figure 4: The phase shifts of $[\Theta_0 + \Phi]$ (left panel) and of $\Pi$ (right panel) induced by different physical effects measured at the LSS.
• Figure 5: Illustration of the projection from three to two dimensions. The round circle is the LSS, the vertical solid(dashed) lines are the peaks(troughs) of mode $\vec{k}$ at $\eta_\star$. The wiggling curve around the LSS is a Legendre polynomial $P_\ell(\mu)$ with $\ell=k(\eta_0-\eta_\star)$, where $\mu = \hat{k} \cdot \hat{\gamma}$, is the cosine of the angle subtend by the wavevector $\vec{k}$ and the direction of observation $\hat{\gamma}$. In the $\mu=0$ direction, the peak-trough separation of the $\vec{k}$ mode matches that of $P_{\ell_m}(\mu)|_{\ell_m =k(\eta_0-\eta_\star)}$ (shown). In the $\mu=1$ direction, the peak-trough separation of the $\vec{k}$ mode matches that of $P_{\ell_m}(\mu)|_{\ell_m \ll k(\eta_0-\eta_\star)}$ (not shown).