First Detection of the Acoustic Oscillation Phase Shift Expected from the Cosmic Neutrino Background

The unimpeded relativistic propagation of cosmological neutrinos prior to recombination of the baryon-photon plasma alters gravitational potentials and therefore the details of the time-dependent gravitational driving of acoustic oscillations. We report here a first detection of the resulting shifts in the temporal phase of the oscillations, which we infer from their signature in the cosmic microwave background temperature power spectrum.

Figure 1

Undamped power spectra ${\mathcal{K}}_l$ [defined in Eq. (1)] with different values of $N_{\nu }$. In all panels the baryon density ${\omega }_b$, the ratio of matter to radiation density ${\rho }_m/({\rho }_{\gamma }+{\rho }_{\nu })$, and the angular size of the sound horizon ${\theta }_s$ are held fixed as these are well determined by CMB data fairly independently of the assumed value of $N_{\nu }$. In the top panel the dominant source of variation is the change in the damping scale ${\theta }_D$ caused by the changes in $N_{\nu }$. In the middle panel we fix ${\theta }_D$ by varying the primordial fraction of baryonic mass in helium appropriately, leaving the dominant source of power spectrum variation as the change in oscillation amplitude $A^{\prime }$. Finally, in the bottom panel, with the spectra normalized to remove the effect of $A^{\prime }$ variation, one can see the subtle impact of the shifts in temporal phase $\varphi$. The data points are the 2013 Planck data.

Figure 2

Posterior reconstructions of the phase shift template $f_l=\alpha \ln (l)+\beta$ under the $\mathrm{\Lambda }\mathrm{CDM}+N_{\nu }+N_{\nu }^{\delta \varphi }$ model, normalized to give the amplitude of the relative shift between the fiducial 3.046 neutrino species and a single specie, along with numerically obtained phase shifts obtained from 100 different simulated cosmologies with varying neutrino species drawn from $N_{\nu }=[1,6]$ relative to the fiducial cosmology, rescaled by the amplitude parameter given in Eq. (3). Also included is the analytic approximation given by Ref. [6], which they also found to be $\sim 25\%$ higher than the numerical result at $\ell \simeq 3000$.

Figure 3

Top: 2D constraints on the jointly varying $\mathrm{\Lambda }\mathrm{CDM}+N_{\nu }+N_{\nu }^{\delta \varphi }$ parameter space. The constraints on $N_{\nu }$ (damping) and $N_{\nu }^{\delta \varphi }$ (phase shift) are essentially orthogonal. Bottom: Constraints from March 2013 planck temperature power spectrum measurements on the number of neutrino species from (1) blue solid line: varying $N_{\nu }^{\delta \varphi }$ while holding $N_{\nu }$ fixed at three and (2) green dashed line: varying along the physical direction $N_{\nu }=N_{\nu }^{\delta \varphi }$. The constraints assume a Gaussian $\tau$ prior of mean $\mu =0.085$ and width $\sigma =0.015$.