The propagating speed of relic gravitational waves and their refractive index during inflation

Abstract

If the refractive index of the tensor modes increases during a conventional inflationary stage of expansion the relic graviton spectrum is tilted towards high frequencies. Two apparently diverse parametrizations of this effect are shown to be related by a rescaling of the fourdimensional metric through a conformal factor that involves the refractive index itself. Non-monotonic spectra with a maximum in the MHz region correspond to a limited variation of the refractive index terminating well before the end of inflation. After exploring a general approach encompassing the ones proposed so far, we estimate the required sensitivity for the direct detection of the predicted gravitational radiation and demonstrate that the allowed regions of the parameter space are within reach for some of the planned detectors operating either in the audio band (like Ligo/Virgo and Kagra) or in the mHz band (like Lisa, Bbo and Decigo).

The refractive index of relic gravitons

“…It is not difficult to show that in this case, as previously discussed, it is impossible to get a large signal in the nHz range without jeopardizing the big-bang nuclosynthesis constraint (see[7,18]). This point is quite well known also from other scenarios where the blue spectra are not induced by the variation of the refractive index[5].…”

“…Other examples include the situation where the higher-order curvature corrections are given in terms of the Gauss-Bonnet combination weighted by some inflaton dependent-coupling [15,16,17]. In [7] (see also [19,18]) it has been argued that in all these situations the effective action of the relic gravitons is be modified and ultimately assumes the following general form:…”

“…( 2) has been proposed in Ref. [7] apparently different parametrizations have been later introduced (see [18,19] and discussion theierin). The difference between these strategies consists in modifying the first term (rather than the second) inside the squared bracket of Eq.…”

The refractive index of the relic gravitons and the nHz band

“…Other examples include the situation where the higher-order curvature corrections are given in terms of the Gauss-Bonnet combination weighted by some inflaton dependent-coupling [16][17][18]. In [8] (see also [19,20]) it has been argued that in all these situations the effective action of the relic gravitons is be modified and ultimately assumes the following general form:…”

Modified gravity as a diagravitational medium

“…As it could be expected, in Minkowski space the effective refractive index of GWs can be characterized by a scalar quantity. It should be noted that a first approach to include modified gravity effects in the scalar refractive index of highly symmetric back-grounds has been recently considered [19,20] during the final stage of development of this work (focusing on cosmological scenarios in that case) .…”