Free-Streaming Neutrinos and Their Phase Shift in Current and Future CMB Power Spectra

TL;DR

This work establishes two complementary template-based methods to detect and quantify the phase shift in CMB acoustic oscillations induced by free-streaming relativistic species, notably the cosmic neutrino background. The spectrum-based template (SBT) shifts the CMB power spectra by a multipole offset and introduces $N_\mathrm{eff}^{\delta\ell}$, while the perturbation-based template (PBT) directly shifts the photon-baryon perturbations via a $k$-dependent phase shift and introduces $N_\mathrm{eff}^{\delta\phi}$. Across Planck, ACT, and SPT data, both approaches detect a nonzero phase shift with high significance and find consistency with the Standard Model value $N_\mathrm{eff}=N_\mathrm{eff}^{\delta\ell}=N_\mathrm{eff}^{\delta\phi}=3.044$, reinforcing three free-streaming neutrino species. The paper also provides forecasts showing that future surveys like Simons Observatory and CMB-S4 will dramatically improve phase-shift constraints, enabling stringent tests for nonstandard neutrino interactions or additional light relics in a signature-driven, model-agnostic framework.

Abstract

The cosmic neutrino background and other light relics leave distinct imprints in the cosmic microwave background anisotropies through their gravitational influence. Since neutrinos decoupled from the primordial plasma about one second after the big bang, they have been free-streaming through the universe. This induced a characteristic phase shift in the acoustic peaks as a unique signature. In this work, we constrain the free-streaming nature of these relativistic species and other light relics beyond the Standard Model of particle physics by establishing two complementary template-based approaches to robustly infer the size of this phase shift from the temperature and polarization power spectra. One template shifts the multipoles in these spectra, while the other novel template more fundamentally isolates the phase shift at the level of the underlying photon-baryon perturbations. Applying these methods to Planck data, we detect the neutrino-induced phase shift at about $10σ$ significance, which rises to roughly $14σ$ with additional data from the Atacama Cosmology Telescope and the South Pole Telescope. We also infer that the data is consistent with the Standard Model prediction of three free-streaming neutrinos. In addition, we forecast the capabilities of future experiments which will enable significantly more precise phase-shift measurements, with the Simons Observatory and CMB-S4 reducing the $1σ$ uncertainties to roughly 4.3% and 2.5%, respectively. More generally, we establish a new analysis pipeline for the phase shift induced by neutrinos and other free-streaming dark radiation which additionally offers new avenues for exploring physics beyond the Standard Model in a signature-driven and model-agnostic way.

Figures (14)

• Figure 1: Illustration of the phase shift as imprinted in the temperature (TT), polarization (EE) and cross-correlation (TE) power spectra with removed Silk damping, $\mathcal{K}_\ell^{XY}$, as defined in \ref{['eq:K_ell']}. To highlight this effect as a function of the energy density of relativistic free-streaming species as parameterized by $N_\mathrm{eff}$, the following quantities were fixed as in Follin:2015hyaWallisch:2018rzj: the physical baryon density $\omega_b$, the scale factor at matter-radiation equality $a_\mathrm{eq}$, the angular size of the sound horizon $\theta_s$, the angular size of the damping scale $\theta_d$ and the height of the fourth peak. The upper panels display the spectra over the observationally most relevant range of multipoles $\ell$ for current CMB experiments, while the lower panels zoom in around the fourth TT peak to better exhibit the imprinted shift. This is the characteristic signature of free-streaming neutrinos and other light relics that we investigate and constrain in this paper.
• Figure 2: Variation of the lensed CMB polarization power spectrum with removed Silk damping, $\mathcal{K}^{EE}_\ell$, defined in \ref{['eq:K_ell']}, as a function of the energy density of relativistic free-streaming species as parameterized by $N_\mathrm{eff}$. To isolate the phase shift, we sequentially fix a number of parameters to their fiducial values. Following Follin:2015hyaWallisch:2018rzj, the physical baryon density $\omega_b$, the scale factor at matter-radiation equality $a_\mathrm{eq}$ and the angular size of the sound horizon at recombination $\theta_s$ are held fixed in all panels. In the top panel, the main effect is the variation of the damping scale $\theta_d$. This is why we fix $\theta_d$ in the second panel by adjusting the primordial helium fraction $Y_p$ which reveals the amplitude shift as a free-streaming signature. In the third panel, we additionally normalize the amplitude of the spectra at the fourth peak such that the remaining variation is the multipole shift $\delta\ell$. The bottom panel presents a smaller multipole range of the third panel to better highlight the shift towards larger angular scales.
• Figure 3: Same as Fig. \ref{['fig:KEE_phase_shift']} for the lensed CMB temperature-polarization cross-spectrum $\mathcal{K}^{TE}_\ell$.
• Figure 4: Template of the spectrum-based multipole shift $f_\ell$ as defined in \ref{['eq:delta_ell_def1']}. The numerical phase shifts for the TT, EE and TE spectra, which are displayed in blue, red and green, respectively, were obtained from sampling 100 different cosmologies with varying number of relativistic species $N_\mathrm{eff} \in [1,6]$ and are normalized to the multipole shift from 3.044 to one species. The error bars indicate the standard deviation in these measurements at the respective peaks/troughs of the CMB power spectra relative to the fiducial cosmology. The best fit for the spectrum-based template $f_\ell$ of \ref{['eq:f_ell_def']} is shown in black and its corresponding $1\sigma$ ($2\sigma$) confidence interval are shaded in dark (light) gold.
• Figure 5: Variation of the perturbations in the photon-baryon fluid at recombination as a function of the amount of free-streaming radiation as parameterized by $N_\mathrm{eff}$. We show the effect on the monopole Sachs-Wolfe term $\Theta_0 + \Psi$ in the top two panels and the polarization field $\Pi$ in the bottom two panels for $N_\mathrm{eff} \in [1, 6]$ as indicated by the colorbar. The second and fourth panels display a much smaller range of wavenumbers for better visualization of the phase shift in these quantities. As for the multipole shift illustrated in Figures \ref{['fig:Kl_phase_shift']}--\ref{['fig:KTE_phase_shift']}, the perturbations are normalized at the fourth peak with $\omega_b$, $a_\mathrm{eq}$, $\theta_s$ and $\theta_d$ held fixed, such that the remaining variation is the phase shift $\delta\phi$.