Long range neutrino forces in the cosmic relic neutrino background

Dispersion potentials arising from double particle exchange have been systematically studied in a wide variety of physical contexts and with quite different scopes and purposes @1#. Indeed, the studies include pure QED phenomena such as van der Waals interactions @2#, two neutrino forces among macroscopic bodies @3#, forces mediated by scalar particles @4,5# found in recent completions of the standard model ~e.g., superlight scalar partners of the gravitino!, etc. In particular two-neutrino exchange forces have been repeatedly scrutinized since first discussed by Feinberg and Sucher. An aspect that has been reanalyzed in recent work @6# is the observation raised in @7# that the cosmic neutrino heat bath has an effect on long range neutrino interactions. In both Refs. @7,6# an approximate neutrino distribution function was used that simplified the calculations. The claim was that for a small neutrino chemical potential, the background neutrinos can be considered nearly Boltzmann distributed and this fact, while only introducing a small distortion into the long range forces, makes the calculations much easier. But the actual phase-space distribution for relic cosmological neutrinos has a Fermi-Dirac shape. Indeed, any fermionic species in thermal equilibrium which at time tD and temperature TD of decoupling was highly relativistic followed an equilibrium distribution n(p,tD)5@exp(E/TD)11# . After decoupling, the energy is redshifted by the expansion of the Universe, E(t)5E(tD)@R(tD)/R(t)# , as the number density decreases like R. As a result, the phase-space distribution at time t will keep the Fermi-Dirac form with the temperature T(t) 5TD@R(tD)/R(t)# . In the present paper we use the exact Fermi-Dirac neutrino distribution function with arbitrary chemical potential and observe that the long distance results are drastically modified even for a small chemical potential in contrast to previous claims. We neglect the effect of a neutrino mass which for the phenomenologically suggested values would not affect the present results. We comment in passing that there has been in the recent literature @8# renewed interest in cosmic neutrino degeneracy which could make neutrino scattering a viable explanation for the ultrahigh-energy cosmic ray events observed so far. We shall adopt the notation in @7,6# and write