Coupled Evolution of Primordial Gravity Waves and Relic Neutrinos

Abstract

We describe analytically the cosmological evolution of primordial gravity waves (tensor perturbations) taking into account their coupling to relic neutrinos. We prove that as a consequence of causality the neutrino-induced phase shift of subhorizon tensor oscillations tends on small scales to zero. For the tensor modes that reenter the horizon in the radiation era after neutrino decoupling we calculate the neutrino suppression factor as 1-5rho_nu/9rho+O[(rho_nu/rho)^2]. This result is consistent with the value obtained for three neutrino flavors by Weinberg and is in agreement with numerical Boltzmann evolution. A minimal formula with the correct asymptotic form on small and large scales reproduces to about ten percent the evolution on all scales probed by the CMB. A more accurate solution (in terms of elementary functions) shows that the modes reentering the horizon in the radiation era are slightly enhanced and the phase of their temporal oscillations is shifted by subdominant nonrelativistic matter. The phase shift grows logarithmically on subhorizon scales until radiation-matter equality; the accumulated shift scales for k>>k_eq as (ln k)/k. The modes reentering the horizon after equality are, in turn, affected by the residual radiation density. These modes follow the naive matter-era evolution which is advanced by a redshift and scale independent increment of conformal time. In an appendix, we introduce a general relativistic measure of radiation intensity that in any gauge obeys a simple transport equation and for decoupled particles is conserved on superhorizon scales for arbitrary initial conditions in the full nonlinear theory.

Figures (5)

• Figure 1: Evolution of tensor metric perturbations in the radiation era as described by a real-space Green's function, left, and Fourier modes, right. For both presentations, the displayed solutions are the neutrinoless, unsourced by any anisotropic stress (dashed) and the linear in $\rho_{\nu}/\rho$ for three neutrino flavors (solid). The Fourier modes on the right panel are additionally compared with the full Boltzmann cmbfast calculation (bold dots).
• Figure 2: Tensor modes at the redshift of CMB last scattering, $z=1090$. The plots are generated with: "minimal" analytical formula (\ref{['trf_gen']}) (solid), numerical cmbfast calculation (bold dots), the neutrino- and adiabatically-damped radiation era solution (dashed), and the matter era solution (dash-dotted).
• Figure 3: The same calculations as in Fig. \ref{['fig_glob']} but for a ten times smaller redshift. "Minimal" analytical expression (\ref{['trf_gen']}) (solid) continues to provide good accuracy.
• Figure 4: The sixth oscillation of a tensor mode with $k=5\tau_{\rm eq}^{-1}$ as: next-to-adiabatic analytical solution (\ref{['h_sub_gen']})-(\ref{['A_sub_radent']})-(\ref{['Df_sub']}) (solid), cmbfast output (dots), and the neutrino- and adiabatically-damped radiation era solution (dashed). The top figure is for the $\rho_{\nu}=0$ model. Both the matter-induced amplitude enhancement and phase shift are seen well reproduced analytically. The bottom figure describes the standard 3-neutrino scenario. The appearing small discrepancy is caused by the interference of the neutrino and matter corrections, as discussed in the main text.
• Figure 5: Evolution of a mode with $k=0.2\,\tau_{\rm eq}^{-1}$ as: the shifted $\tau\to\tau+\tau_e$ matter era solution (solid), the regular matter era solution (dash-dotted), and cmbfast integration (dots).