We saw above that fluctuations in a scalar field during inflation get turned
into temperature fluctuations via the intermediary of gravity. Gravity affects
in more ways than this. The Newtonian potential and spatial curvature
alter the acoustic oscillations by providing a gravitational
force on the oscillator. The Euler equation (8)
gains a term on the rhs due to the gradient of the potential
. The main effect of gravity then is to make the oscillations a
competition between pressure gradients
and potential gradients
with an equilibrium when
.
Gravity also changes the continuity equation. Since the Newtonian curvature
is essentially a perturbation to the scale factor,
changes in its value also generate temperature perturbations by analogy to the
cosmological redshift and so the continuity equation (7)
gains a contribution of
on the rhs.
These two effects bring the oscillator equation (9)
to
![]() |
(14) |
In a flat universe and in the absence of pressure, and
are constant. Also, in the absence of baryons,
so the new oscillator equation is identical to Equation (9)
with
replaced by
. The solution in the matter dominated epoch is then
where represents the start of the matter dominated epoch (see
Figure 1a). We have used the matter dominated
``initial conditions'' for
given in the previous section assuming large scales,
.
The results from the idealization of §3.1
carry through with a few exceptions. Even without an initial temperature fluctuation
to displace the oscillator, acoustic oscillations would arise by the infall
and compression of the fluid into gravitational potential wells. Since it is
the effective temperature that oscillates, they occur even if
. The quantity
can be thought of as an effective temperature in another
way: after recombination, photons must climb out of the potential well to the
observer and thus suffer a gravitational redshift of
. The effective temperature fluctuation is therefore
also the observed temperature fluctuation. We now see that the large scale limit
of Equation (15) recovers the
famous Sachs-Wolfe result that the observed temperature perturbation is
and overdense regions correspond to cold spots on the sky [Sachs & Wolfe, 1967]. When
, although
is positive, the effective temperature
is negative. The plasma begins effectively rarefied in gravitational
potential wells. As gravity compresses the fluid and pressure resists, rarefaction
becomes compression and rarefaction again. The first peak corresponds to the
mode that is caught in its first compression by recombination. The second peak
at roughly half the wavelength corresponds to the mode that went through a full
cycle of compression and rarefaction by recombination. We will use this language
of the compression and rarefaction phase inside initially overdense regions
but one should bear in mind that there are an equal number of initially underdense
regions with the opposite phase.