Kiefer C. Quantum gravity

INTRODUCTION OF INHOMOGENEITIES 259
to be small perturbations of the homogeneous background described by a and
the homogeneous field φ. Since one knows from measurements of the microwave
background radiation that the fluctuations were small in the early universe, this
approximation may be appropriate for that phase. The multipole expansion is
also needed for the description of decoherence (Section 10.1.2). Cosmological
perturbations were first studied by Lifshits (1946).
Following Halliwell and Hawking (1985), we make for the three-metric the
ansatz
h ab = a 2 (Ω ab + ɛ ab ) , (8.36)
where Ω ab denotes the metric on S 3 , and the ‘perturbation’ ɛ ab (x,t) is expanded
into spherical harmonics,
ɛ ab (x,t)= ∑ (√
2
3 a n(t)Ω ab Q n + √ 6b n (t)Pab n + √ )
2c n (t)Sab n +2d n(t)G n ab .
{n}
(8.37)
Here {n} stands for the three quantum numbers {n, l, m}, wheren =1, 2, 3,...,
l =0,...,n− 1, and m = −l,...,l. The scalar field is expanded as
Φ(x,t)= √ 1 φ(t)+ɛ(x,t) ,
2π
ɛ(x,t)= ∑ {n}
f n (t)Q n . (8.38)
The scalar harmonic functions Q n ≡ Q n lm on S3 are the eigenfunctions of the
Laplace operator on S 3 ,
Q n |k
lm|k = −(n2 − 1)Q n lm , (8.39)
where |k denotes the covariant derivative with respect to Ω ab . The harmonics
can be expressed as
Q n lm(χ, θ, φ) =Π n l (χ)Y lm (θ, φ) , (8.40)
where Π n l (χ) are the ‘Fock harmonics’, and Y lm(θ, φ) are the standard spherical
harmonics on S 2 . They are orthonormalized according to
∫
dµQ n lm Qn′ l ′ m = ′ δnn′ δ ll ′δ mm ′ , (8.41)
S 3
where dµ =sin 2 χ sin θdχdθdϕ. The scalar harmonics are thus a complete orthonormal
basis with respect to which each scalar field on S 3 can be expanded.
The remaining harmonics appearing in (8.37) are called tensorial harmonics of
scalar type (Pab n ), vector type (Sn ab ), and tensor type (Gn ab
); see Halliwell and
Hawking (1985) and the references therein. At the present order of approximation
for the higher multipoles (up to quadratic order in the action), the vector
harmonics are pure gauge. The tensor harmonics G n ab
describe gravitational