40 An introduction to the physics of cosmology
slow-roll condition, since manipulation of these three equations is often required
in derivations.
2.5.4 Gravity waves and tilt
The density perturbations left behind as a residue of the quantum fluctuations in
the inflaton field during inflation are an important relic of that epoch, but are not
the only one. In principle, a further important test of the inflationary model is
that it also predicts a background of gravitational waves, whose properties couple
with those of the density fluctuations.
It is easy to see in principle how such waves arise. In linear theory, any
quantum field is expanded in a similar way into a sum of oscillators with the
usual creation and annihilation operators; this analysis of quantum fluctuations in
a scalar field is thus readily adapted to show that analogous fluctuations will be
generated in other fields during inflation. In fact, the linearized contribution of a
gravity wave,hμν, to the Lagrangian looks like a scalar fieldφ=(mP/ 4
√
π)hμν,
the expected rms gravity-wave amplitude is
hrms∼H/mP.The fluctuations inφare transmuted into density fluctuations, but gravity waves
will survive to the present day, albeit redshifted.
This redshifting produces a break in the spectrum of waves. Prior to horizon
entry, the gravity waves produce a scale-invariant spectrum of metric distortions,
with amplitudehrmsper lnk. These distortions are observable via the large-scale
CMB anisotropies, where the tensor modes produce a spectrum with the same
scale dependence as the Sachs–Wolfe gravitational redshift from scalar metric
perturbations. In the scalar case, we haveδT/T ∼ φ/ 3 c^2 ,i.e.oforderthe
Newtonian metric perturbation; similarly, the tensor effect is
(
δT
T)
GW∼hrms.δH∼ 10 −^5 ,where the second step follows because the tensor modes can constitute no more
than 100% of the observed CMB anisotropy.
A detailed estimate of the ratio between the tensor effect of gravity waves
and the normal scalar Sachs–Wolfe effect was first analysed in a prescient paper
by Starobinsky (1985). Denote the fractional temperature variance per natural
logarithm of angular wavenumber by^2 (constant for a scale-invariant spectrum).
The tensor and scalar contributions are, respectively,
^2 T∼h^2 rms∼(H^2 /m^2 P)∼V/m^4 P^2 S∼δH^2 ∼H^2
φ ̇ ∼H^6
(V′)^2
∼
V^3
m^6 PV′^2