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 \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 initial conditions can not 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 to 0.09 , depending on the constraints on the helium abundance .