Introduction to Cosmology

Density Fluctuations 225

wherekis the wave vector in comoving coordinates. An arbitrary pattern of fluctua-
tions can be described mathematically by an infinite sum of independent waves, each
with its characteristic wavelength휆or comoving wavenumber푘and its amplitude훿푘.
The sum can be formally expressed as a Fourier expansion for the density contrast

훿m(r,푡)∝

∑

훿푘(푡)eik⋅r. (10.6)

A density fluctuation can also be expressed in terms of the mass푀moved within
one wavelength, or rather within a sphere of radius휆, thus푀∝휆^3. It follows that the
wavenumber or spatial frequency푘depends on mass as

푘=^2 휋

휆

∝푀−^1 ∕^3. (10.7)

Power Spectrum. The density fluctuations can be specified by the amplitudes훿푘of
the dimensionlessmass autocorrelation function

휉(푟)=⟨훿(r 1 )훿(r+r 1 )⟩∝

∑

⟨|훿푘(푡)|^2 ⟩eik⋅r, (10.8)

which measures the correlation between the density contrasts at two pointsrandr 1.
The powers|훿푘|^2 define the power spectrum of the root-mean-squared (RMS) mass
fluctuations

푃(푘)=⟨|훿푘(푡)|^2 ⟩. (10.9)

Thus the autocorrelation function휉(푟)is the Fourier transform of the power spectrum.
We have already met a similar situation in the context of CMB anisotropies, where the
waves represented temperature fluctuations on the surface of the surrounding sky.
There Equation (8.18) defined the autocorrelation function퐶(휃)and the powers푎^2 퓁
were coefficients in the Legendre polynomial expansion Equation (8.19).
Taking the power spectrum to be of the phenomenological form [Equation (7.60)],

푃(푘)∝푘푛,

and combining with Equation (10.7), one sees that each mode훿푘is proportional to
some power of the characteristic mass enclosed,푀훼.
Inflationary models predict that the mass density contrast obeys

훿^2 m∝푘^3 ⟨|훿푘(푡)|^2 ⟩ (10.10)

and that the primordial fluctuations have approximately a Harrison–Zel’dovich spec-
trum with푛s=1. Support for these predictions come from the CMB temperature
and polarization asymmetry spectra which give the value quoted in Equation (8.47),
푛= 0. 960 ± 0 .0073 [1].
Independent, although less accurate information about the spectral index can be
derived from constraints set by CMB isotropy, galaxies and black holes using훿푘∝푀훼.
The CMB scale (within the size of the present horizon of푀≈ 1022 푀⊙) is isotropic to
less than about 10−^4. Galaxy formation (scale roughly 10^12 푀⊙) requires perturbations
of order 10−^4 ±^1. Taking the ratio of perturbations versus mass implies a constraint on
훼and implies that푛is close to 1.0 at long wavelengths.