Relic Gravitons, Dominant Energy Condition and Bulk Viscous Stresses

If the energy momentum tensor contains bulk viscous stresses violating the dominant energy condition (DOC) the energy spectra of the relic gravitons (produced at the time of the DOC’s violation) increase in frequency in a calculable way. In a general relativistic context we give examples where the DOC is only violated for a limited amount of time after which the ordinary (radiation dominated) evolution takes place. We connect our discussion to some recent remarks of Grishchuk concerning the detectability of the stochastic gravitational wave background by the forthcoming interferometric detectors. Electronic address: [email protected] Every transition of the background geometry leads, inevitably, to the production of stochastically distributed relic gravitons whose energy spectra represent a crucial probe of the very early stages of the evolution of our Universe [1]. Prior to the nucleosynthesis epoch there are no direct test of the thermodynamical state of the Universe and the presence of an inflationary phase of expansion is usually justified by causality arguments applied to the Cosmic Microwave Background (CMB) photons whose emission regions could not have been in causal contact in the far past if a never ending radiation dominated phase would precede our present matter dominated stage of expansion [2]. An inflationary evolution, if regarded at an effective level, violates the (general relativistic) strong energy condition (SEC) namely ρ + 3p ≥ 0 where ρ and p are, respectively, the energy density and the pressure density of the perfect fluid sources driving the evolution of the background geometry [3]. The need for such a violation can be immediately seen from the structure of Friedmann equations which imply that, if ä > 0, the SEC needs to be violated (a(t) is the scale factor of the homogeneous and isotropic Universe and the over-dot denotes the differentiation with respect to the cosmic time t). Recently, Grishchuk [4, 5] made an interesting observation concerning the slopes of the energy spectra of relic gravitons emerging from the models of the early Universe. In short the argument goes as follows. Suppose that we are in the framework of general relativity and suppose, as it is usually done, that the effective sources driving the background geometry can be parameterized by a stress tensor with perfect fluid form. As we stressed at the beginning, there are little doubts that the Universe had to be dominated by radiation (at least) since nucleosynthesis. Prior to that epoch we do not know which kind of energy momentum tensor could approximate the background sources but we would like to deal with ever expanding Universes. Moreover, at least for some time, we would like an accelerated expansion in order to solve the so called kinematical problems of the standard cosmological model. The second of the two previous requirements necessarily leads to the violation of the SEC. A very naive model of the early Universe would then be given by two phases. An unknown phase where the perfect fluid sources have a generic equation of state p = γρ , followed, at some transition time t1, by a radiation dominated phase with p = ρ/3. The question we are very interested to ask is under which conditions such a toy model would produce energy spectra of relic gravitons increasing faster than the first power of the frequency. Such a question is of obvious experimental relevance since, in the near future, various interferometric detectors of gravitational waves will come in operation. Now, if the energy spectra of relic gravitons are flat (or slowly increasing) with frequency there are little hopes of detecting them. In fact, the COBE limit applied to a frequency ν0 ∼ 10 h0 Hz imposes that the relic gravitons energy density (in critical units) h0ΩGW has to be smaller then (or of the order of) 6.9× 10 (h0 is the present indetermination in the value of the Hubble constant). Due to the transition from radiation to matter, the infra-red branch of the graviton spectrum declines (in frequency) as ν between ν0 and ν2 ∼ 10 Ω0 h0 Hz. If we take into account that the typical frequency of operation of the interferometers is between 10 and 100 Hz,