^{3}.

Damping of primordial gravitational waves due to the anisotropic stress contribution owing to the cosmological neutrino background (C

The theoretical investigation and the phenomenological analysis of anisotropies in the cosmological microwave background (C

Cosmological tensor fluctuations should produce not only temperature anisotropies but also distinct imprints in the so-called magnetic or

The observed pattern of temperature anisotropies, when combined with probes of inhomogeneities in matter on large scale structures, and with measurements of the total energy density in the universe, is in striking agreement with the simplest predictions for the spectrum of anisotropies due to gravitational waves produced during inflation. These facts support the inclusion of extra ingredients in the fine-tuning analysis involving the theoretical predictions and the observable data for C

The C

Our aim is to extend such a preliminary approach involving the RD cosmic inventory to a transient, radiation-to-matter dominated (RMD) background universe. We will follow the analytical setup based on the multipole formalism that reproduces the procedure which deals with scalar perturbations [

Our paper is therefore organized as follows. In Section

As supported by the decomposition theorem [

The propagation of gravitational waves is parameterized by the relevant (spatial) components of the perturbed metric written as (one notices the (

By turning (

The analytical multipole decomposition discussed in [

Finally, by multiplying (

Since we are concerned with the fact that the anisotropic stress is only cosmologically relevant for massless particles [

The background solutions of the Friedmann equation for the RMD universe, with the corresponding equation of state, respectively, represented by

The system of coupled equations ordinarily defined in terms of

The corresponding dynamical evolution of the gravitational waves, that is, of the tensor modes,

Evolution of the normalized wave amplitude

The neutrino free-streaming regime is obtained by setting a vanishing collision term,

The inclusion of the matter background into the cosmic inventory introduces an additional subtle effect on the amplitude of the gravitational waves under the influence of the anisotropic stress. Scales just entering the horizon at late times have the corresponding oscillation modes undergoing a delayed suppression due to the coupling to neutrinos. It propagates to the following oscillation peaks in a kind of translational effect of the oscillation pattern, which is naturally expected if one observes that, in the limit of radiation domination, one has

In [

Time-averaged values of

Evolution of the normalized wave amplitude

When the elements for describing the RMD regime are introduced, our results are considerably different, in spite of exhibiting a conceptual agreement with those from [

The maximum amount of damping occurs for the extrapolating limit of

Finally, the highest first oscillation peak for the RMD results depicted in Figure

Turning back to the collision term contributions, we will follow the parametrization from [

Time-averaged values of

Figures

Figure

To end up, the damping of gravitational waves also affects some spectral features related to the tensor contribution to the anisotropy spectrum. From the analytical multipole decomposition [

Substituting the results for

The numerical results obtained for the tensor modes,

After entering the horizon, the amplitude of gravitational waves dies away (cf. Figure

Tensor contribution to the angular power spectrum in the RMD scenario. Results are for vanishing anisotropic stress (dotted lines), for

Tensor contribution to the angular power spectrum for the RMD scenario in case of including the collision terms. Results are for

Figure

The observation of primordial gravitational waves indeed provides a renewed overview about the earliest moments in the history of the universe and on possible new physics at energies many orders of magnitude beyond those accessible at particle accelerators. The recent positive fit-back from experimental physics [

Since a universe overfilling viscosity results in gravitational wave damping effects, we have considered the possibility of observing some frequency-dependent absorption in the frequency range where neutrino decoupling is relevant. By mixing analytical and numerical procedures, we have obtained the evolution of tensor modes and its corresponding imprints on the C

The damping effects owing to the influence of the (neutrino) anisotropic stress have been computed for a RMD scenario and compared to previous results for the RD scenario. We have compared the effects of including collision terms with collision frequency parameterized by

The connection between the anisotropic stress of neutrinos and its effects on the C

Finally, a time-averaged quantity,

Figures

Evolution of the normalized wave amplitude

Evolution of the normalized wave amplitude

The authors declare that there is no conflict of interests regarding the publication of this paper.

A. E. Bernardini would like to thank the Brazilian Agencies FAPESP (Grant 08/50671-0) and CNPq (Grant 300233/2010-8) for their financial support.