Physical Foundations of Cosmology

9.5 Anisotropies on large angular scales 367

IfδT/Tis expanded in terms of spherical harmonics,

δT(θ,φ)

T 0

=

∑

l,m

almYlm(θ, φ), (9.39)

then the complex coefficientsalm,in a homogeneous and isotropic universe, satisfy
the condition
〈al∗′m′alm〉=δll′δmm′Cl, (9.40)
where the brackets refer to a cosmic mean. The multipole moments,Cl=〈|alm|^2 〉,
receive their main contribution from fluctuations on angular scaleθ∼π/land
l(l+ 1 )Clis about typical squared temperature fluctuations on this scale.

Problem 9.3Generalize the formula (9.38) for ′k(ηr)=0, thereby incorporating

the integrated Sachs–Wolfe effect.

Inflation predicts a flat universe with a nearly scale-invariant, adiabatic spectrum

of Gaussian fluctuations. As we shall show, these lead to certainqualitativefeatures

in the temperature anisotropy power spectrum: a flat plateau for large angular scales,

a sequence of peaks and valleys with a first peak atl≈200, and a steady damping

of the oscillation amplitude aslincreases. Once these features are confirmed, then

a precise measurement of the power spectrum can be used to constrain many of

the cosmological parameters which inflation does not fix uniquely. First, there are

the amplitudeBand spectral indexnof the primordial density inhomogeneities

generated by inflation. The rather generic prediction of inflation is that| (^2) kk^3 |=
BknS−^1 ,with 1−nS∼0.03–0.08. The amplitudeBis not predicted by inflation. Its
value is chosen to fit the observations. The other parameters involved in defining the
shape of the temperature power spectrum are the Hubble constanth 75 ,the fraction
of the critical density today due to the baryon density (^) b, the total matter (baryonic
plus cold dark matter) density (^) mand the vacuum (or quintessence) energy density
(^) .
The present data are consistent with inflation, and suggest a flat universe that
consists approximately of 5% baryonic matter, 25% cold dark matter, and 70% dark
energy. We will take these values for our fiducial model, also called theconcordance
model, and compute the temperature fluctuation spectrum for a range of parameters
around this model.

9.5 Anisotropies on large angular scales

The fluctuations on large angular scales (θ 1 ◦) are induced by inhomogeneities

with wavelengths which exceed the Hubble radius at recombination and have not

had a chance to evolve significantly since the end of inflation. Thus their spectrum