Signatures of relativistic neutrinos in CMB anisotropy and matter clustering

We present a detailed analytical study of ultrarelativistic neutrinos in cosmological perturbation theory and of the observable signatures of inhomogeneities in the cosmic neutrino background. We note that 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 the analytical approach can be extended beyond fluid models to neutrinos. The faster cosmological expansion due to the neutrino background affects 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 $l\gtrsim 200$ 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 modes, 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 $\sigma {(N}_{\nu })\simeq 0.05–0.09,$ depending on the constraints on the helium abundance.