(ed.). Gravitational waves (IOP, 2001)(422s).

Introduction 183contribution to the energy density from a single species of relativistic particlewith g i internal states (helicity, colour, etc) is g i (π 2 /30)T 4 for a boson and(7/8)g i (π 2 /30)T 4 for a fermion. Taking into account that the ith species hasin general a temperature T i ̸= T if it already dropped out of equilibrium, we candefine a function g(T ) from ρ rad = (π 2 /30)g(T )T 4 . Then [1]g(T ) =∑i=bosonsg i(TiT) 4+ 7 8∑i=fermions( ) 4 Tig i (12.2)Twhere the sum runs over relativistic species. This holds if a species is in thermalequilibrium at temperature T i . If instead it does not have a thermal spectrum(which in general is the case for gravitons) we can still use the above equation,where for this species T i does not represent a temperature but is defined (forbosons) from ρ i = g i (π 2 /30)Ti 4,where ρ i is the energy density of this species.The quantity g S (T ) used before for the entropy is given by the same expressionas g(T ), with (T i /T ) 4 replaced by (T i /T ) 3 . We see that both g(T ) and g S (T )give a measure of the effective number of species. For most of the early historyof the universe, g(T ) = g S (T ), and in the standard model at T ² 300 GeV theyhave the common value g ∗ = 106.75, while today g 0 = 3.36 [1]. ThereforeH∗ 2 = 4π 3 g ∗ T∗445MPl2 , (12.3)and, using f ∗ ≡ H ∗ /ɛ, equation (12.1) can be written as [2]f 0 ≃ 1.65 × 10 −7 1 ( )T∗(g∗) 1/6Hz. (12.4)ɛ 1 GeV 100This simple equation allows us to understand a number of important pointsconcerning the energy scales that can be probed in GW experiments. The simplestestimate of f ∗ corresponds to taking ɛ = 1 in equation (12.4) [4]. In this case,we would find that a graviton observed today at the frequency f 0 = 100 Hz, thescale relevant for VIRGO, was produced when the universe had a temperatureT ∗ ∼ 6 × 10 8 GeV. Since in the RD phase one has [1]t = 1 ( )2H = 45 1/2M Pl16π 3 g(T ) T 2 ≃ 2.42 ( ) MeV 2g 1/2 s, (12.5)Tthis temperature corresponds to a production time t ∗ ∼ 7 × 10 −25 s. At this timethe graviton had an energy E ∗ ∼ 3 GeV.However, the fact that it makes sense to consider gravitons production onlyfor wavelengths with λ ∗ º H∗−1 does not necessarily mean, in general, that att = t ∗ the typical wavelength of GWs produced will be at λ ∗ ∼ H∗−1 (ɛ ∼ 1) evenas an order of magnitude estimate. In [5] this point is illustrated with two specificexamples, one in which the assumption λ ∗ ∼ H∗−1 turns out to be basically