Introduction to Cosmology

Temperature Anisotropies 183

Curvature and isocurvature fluctuations behave differently when they are super-
horizon: isocurvature perturbations cannot grow, while curvature perturbations can.
Once an isocurvature mode passes within the horizon, however, local pressure
can move energy density and can convert an isocurvature fluctuation into a true
energy-density perturbation. For subhorizon modes the distinction becomes unim-
portant and the Newtonian analysis applies to both. However, isocurvature fluctua-
tions do not lead to the observed acoustic oscillations seen in Figure 8.3 (they do not
peak in the right place), whereas the adiabatic picture is well confirmed.
At the LSS, crests in the matter density waves imply higher gravitational potential.
As we learned in Section 2.5, photons ‘climbing out’ of overdense regions will be red-
shifted by an amount given by Equation (3.1), but this is partially offset by the higher
radiation temperature in them. This source of anisotropy is called theSachs–Wolfe
effect. Inversely, photons emitted from regions of low density ‘roll down’ from the
gravitational potential, and are blueshifted. In the long passage to us they may tra-
verse further regions of gravitational fluctuations, but then their frequency shift upon
entering the potential is compensated for by an opposite frequency shift when leaving
it (unless the Hubble expansion causes the potential to change during the traverse).
They also suffer a time dilation, so one effectively sees them at a different time than
unshifted photons. Thus the CMB photons preserve a ‘memory’ of the density fluctua-
tions at emission, manifested today as temperature variations at large angular scales.
An anisotropy of the CMB of the order of훿푇∕푇≈ 10 −^5 is, by the Sachs–Wolfe effect,
related to a mass perturbation of the order of훿≈ 10 −^4 when averaged within one
Hubble radius.

90 °

6000

5000

4000

3000

D

[μℓ

(^2) K
]
2000
1000
0
2 10 50 500 1000
Multipole moment, ℓ
1500 2000 2500
18 ° 1 ° 0.2°
Angular scale
0.1° 0.07°
Figure 8.3 The best-fit power spectra of CMB temperature (T) fluctuations as a function
of angular scale (topxaxis) and multipole moment (bottomxaxis) [6]. Reproduced from
the freely accessible Planck Legacy Archive with permission of Jan Tauber, European Space
Agency. (See plate section for color version.)