Signatures of Relativistic Neutrinos in CMB Anisotropy and Matter Clustering

TL;DR

This work develops an analytic, real-space perturbation framework for ultra-relativistic neutrinos in a radiation–matter universe, introducing coordinate-number overdensities that remain conserved on superhorizon scales and using Green's functions to capture neutrino free streaming. The analysis reveals a unique additive phase shift in CMB acoustic oscillations induced by neutrino perturbations, alongside modest suppression of small-scale CMB power and enhanced matter clustering on those scales. These neutrino signatures depend on the effective number of relativistic species Nν and interact with helium abundance Y, leading to degeneracies that polarization measurements can help break. Forecasts for Planck, ACT, and a future CMB polarization mission indicate that Nν can be constrained to σ(Nν) ~ 0.24 (Planck), ~0.09 (with polarization), or ~0.05 (with Y constrained), highlighting the potential to test BBN consistency and nonstandard radiation content with upcoming data.

Abstract

We present a detailed analytical study of ultra-relativistic neutrinos in cosmological perturbation theory and of the observable signatures of inhomogeneities in the cosmic neutrino background. We note that a modification of perturbation variables that removes all the time derivatives of scalar gravitational potentials from the dynamical equations simplifies their solution notably. The used perturbations of particle number per coordinate, not proper, volume are generally constant on superhorizon scales. In real space an analytical analysis can be extended beyond fluids to neutrinos. The faster cosmological expansion due to the neutrino background changes the acoustic and damping angular scales of the cosmic microwave background (CMB). But we find that equivalent changes can be produced by varying other standard parameters, including the primordial helium abundance. The low-l integrated Sachs-Wolfe effect is also not sensitive to neutrinos. However, the gravity of neutrino perturbations suppresses the CMB acoustic peaks for the multipoles with l>~200 while it enhances the amplitude of matter fluctuations on these scales. In addition, the perturbations of relativistic neutrinos generate a *unique phase shift* of the CMB acoustic oscillations that for adiabatic initial conditions cannot be caused by any other standard physics. The origin of the shift is traced to neutrino free-streaming velocity exceeding the sound speed of the photon-baryon plasma. We find that from a high resolution, low noise instrument such as CMBPOL the effective number of light neutrino species can be determined with an accuracy of sigma(N_nu) = 0.05 to 0.09, depending on the constraints on the helium abundance.

Figures (7)

• Figure 1: The evolution of superhorizon adiabatic perturbations in the radiation-matter universe. a) Neutrino anisotropic stress potential. b) The Newtonian gauge gravitational potentials. On both plots, the solid curves show the full result for 3 neutrino species. The dotted curves correspond to 0 neutrino species. The dashed curves on plot b) are the sums of the leading and subleading terms in the expansion of the potentials in $R_{\nu}$, for 3 neutrino species. The dashed vertical lines show $a/a_{\rm eq}$ at CMB decoupling, given the cosmological parameters of Ref. WMAPGen.
• Figure 2: The functions $f_{\pi}$, $f_{\Phi}$, $f_{\Psi}$, and the combination $2f_{\Phi}-f_{\Psi}$ that appear in the $O(R_{\nu})$ order of superhorizon perturbation evolution, as considered in the main text. The evolution variable $r$ is defined by eq. (\ref{['r_def']}). The radiation density domination corresponds to $r\to1$, the matter density domination to $r\to0$.
• Figure 3: a) Adiabatic Green's functions for neutrino (solid) and photon (dashed) number density perturbations in the radiation era. The neutrino fraction, $R_{\nu}$, of the radiation density is assumed infinitesimal. b) Adiabatic Green's functions for the gravitational potentials $\Phi_{\pm}\equiv(\Psi\pm\Phi)/2$ in the radiation era. The solid and dashed curves are the sums of the $O(R_{\nu}^0)$ and $O(R_{\nu})$ terms for three neutrino species. The dotted line is $\Phi_+=\Phi$ for $R_{\nu}\to0$.
• Figure 4: a) Numerically calculated photon number density perturbation $d_{\gamma}$ in the radiation era for 0 and 3 neutrino species $N_{\nu}$ (dots) versus the theoretical prediction (\ref{['d_g_rad_sol_k']}) for $N_{\nu}=0$ (dashed) and its rescaled and phase shifted asymptotic form (\ref{['dg_trf_cor']}, \ref{['Aph_res']}) for $N_{\nu}=3$ (solid). b) Similar comparison for the dark matter density perturbation $d_c$ and $N_{\nu}=0,1,3$. The theoretical predictions are given by eqs. (\ref{['dc_trf0']}--\ref{['Ac_res']}). In all the cases the $O(R_{\nu}^2)$ terms in the analytical formulas are neglected.
• Figure 5: The relative change in $C_l^{TT}$ (top left), $C_l^{EE}$ (top right), matter density perturbation $\delta_m(k/h)$ (bottom left), and $C_l^{\kappa \kappa}$ (bottom right) when $N_{\nu}$ varies from 2.5 to 3.5. The solid curves show the changes when all the other parameters, listed in Sec. \ref{['sec_forecasts']}, are fixed. The top two panels also show the change for fixed recombination history and equivalent Silk damping (dotted, blue). The dotted, blue curve on the bottom left panel gives the change in $\delta_m$ when $\omega_b/\omega_m$ is held fixed.