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Gravitons are the propagating modes associated with transverse, traceless tensor metric perturbations, and they behave as a superposition of two minimally coupled scalar fields, each corresponding to a polarisation state. As a result, the graviton field, which is massless, has a spectrum of quantum mechanical fluctuations similar to the one obtained for the scalar field phi. For each polarisation state, the rms amplitude of the tensor metric perturbations associated with a given Fourier mode at horizon crossing is then (48)

Equation 10

where V is the volume associated with the Fourier expansion, and the value of H is to be calculated when the comoving scale k crosses outside the Hubble radius during inflation. The phases of the Fourier modes hk are again independent and randomly distributed. Consequently, the power spectrum at horizon re-entry, Pt(k), contains all the information necessary to describe the stochastic background of gravitational waves generated during inflation. Again, in general, Pt(k) can be approximated as a single power-law,

Equation 11

A scale-invariant power spectrum for the tensor perturbations then corresponds to nt = 0. For the simplest single-field inflationary models, as long as the gravitational waves are produced during the slow-roll phase, necessarily nt leq 0. Further, under such conditions, and in the simplest models, the ratio between Pg(k) and deltaH2(k) is approximately equal to - nt/2, which is another way of stating the so-called inflationary consistency relation (56). Note that in these models, if the expansion of the Universe during inflation is perfectly exponential, one gets nt = 0, but then the amplitude of the tensor perturbation also effectively tends to zero compared with the amplitude of the (equally scale-invariant) scalar perturbation. However, in some slow-roll models with more complicated potentials, like intermediate inflation, it is possible to have an important tensor contribution to the perturbation spectrum at horizon re-entry though the power spectrum of density perturbations is scale-invariant.

If a gravitational wave background produced during inflation was detected using the CMBR, and it was possible to estimate both the amplitude and spectral index of the tensor perturbation contribution to the CMBR temperature anisotropy and polarisation signals, the next step would be to try to detect locally those gravitational waves. If successful, such a detection would do much the same for the credibility of inflation as was achieved for the Hot Big Bang theory itself by the detection of the CMBR.

The simplest quantity to compare with the experimental sensitivity of gravitational wave detectors is the present-day contribution per logarithmic interval of the gravitational waves to the total energy density,

Equation 12

where rhoc = 3H02 / 8piG and rhogw is the energy density of the stochastic background of gravitational waves with comoving wavenumber k. Given that

Equation 13

we then get in the case of a initial scale-invariant power spectrum of gravitational waves

Equation 14

The shape of the initial power spectrum is broken at the scale of matter-radiation equality, keq = 6.22 x 10-2 Omega0h2 sqrt(3.36/g*) Mpc-1, where g* is the effective number of relativistic degrees of freedom (equals 3.36 for the standard cosmology with 3 massless neutrino species), as gravitational waves that enter the horizon prior to matter-radiation equality redshift more slowly with time. Given that we are only interested in gravitational waves that can be detected locally, we will concentrate on those which entered the horizon during radiation domination. These correspond to k << 2 x 10-24m, or f << 10-16 Hz, where f = c k / 2pi is the frequency.

Working within the simplest inflationary models, those which obey the consistency relation T/S appeq - 7nt, and requiring the CMBR temperature anisotropies detected on large-angular scales by COBE to be reproduced, one obtains (75)

Equation 15

If the power spectrum of tensor perturbations is not a perfect power-law there will be small corrections to this expression, which will be in principle more important for the smallest scales, i.e. high k. A sensitivity of Omegagw(k) h2 ~ 10-15 is therefore needed for a serious search for local gravitational waves produced during inflation. With its initial strain detectors, the Earth-based LIGO (Laser Interferometer Gravitational Wave Observatory) should be able to identify a stochastic background of gravitational waves provided Omegagwh2 is at least a few times 10-3, at its most sensitive operating frequency of roughly 100 Hz, with the limit dramatically improving by possibly 6 orders of magnitude with more advanced strain detectors installed in a later phase (59). Unfortunately, this misses the mark by six orders of magnitude.

Because the energy density in gravitational waves is proportional to the rms strain Deltah squared times frequency squared, a detector operating at lower frequency has better energy-density sensitivity for fixed strain sensitivity. Earth-based detectors cannot operate at frequencies below about 10 Hz because of seismic noise, but space-based operators can. The LISA (Laser Interferometer Space Antenna) mission has already been approved by ESA, with an initialy predicted launch date for only around 2020, but which may be brought down to later this decade if NASA gets interested in a joint effort. It will have a peak sensitivity in terms of Omegagwh2 of about 10-12 at a frequency close to 10-3 Hz (59), which is more promising, but still misses by at least three orders of magnitude the required sensitivity level for the detection of a local stochastic background of inflationary produced gravitational waves. Also, at frequencies above 10-4 Hz it is expected that the stochastic background of gravitational waves produced by compact white-dwarf binaries will swamp the inflationary signal. However, LISA may be able to disentangle the two backgrounds, given that it rotates in orbit and so it will be sensitive at different times to regions in the Galactic plane, where the binaries are, and outside (59).

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