where
R
describes the size of the Universe,
G
is the gravitational constant,
is the present density of the Universe,
k
is a measure of the curvature of space and
is the Cosmological constant which can be thought of as the zero
energy of a vacuum. If the Cosmological term dominates (as the
scalar field is expected to at very high temperatures), then the
other two terms become negligible and it is possible to solve and
find
Therefore, the inflationary theory describes an exponential
expansion of space in the very early Universe. Amplification of
initial quantum irregularities then results in a spectrum of long
wavelength perturbations on scales initially bigger than the
horizon size. Central to the theory of inflation is the potential
, which describes the self-interaction of the scalar inflaton
field
. Due to the unknown nature of this potential, and the unknown
parameters involved in the theory, inflationary theory is bad (at
the moment) at predicting the overall amplitude of the matter
fluctuations at recombination. However, there is a reasonable
agreement that the
form
of the fluctuation spectrum coming out of inflation should be
given by
where k is the comoving wavenumber and n is the `tilt' of the primordial spectrum. The latter is predicted to lie close to 1 (the case n =1 being the Harrison-Zeldovich, or `scale-invariant' spectrum).
An overdensity in the early Universe does not collapse under the effect of self-gravity until it enters its own particle horizon when every point within it is in causal contact with every other point. The perturbation will continue to collapse until it reaches the Jean's length, at which time radiation pressure will oppose gravity and set up acoustic oscillations. Since overdensities of the same size will pass the horizon size at the same time, they will be oscillating in phase. These acoustic oscillations occur in both the matter field and the photon field and so will induce `Doppler peaks' in the photon spectrum.
The level of the Doppler peaks in the power spectrum depend on
the number of acoustic oscillations that have taken place since
entering the horizon. For overdensities that have undergone half
an oscillation, there will be a large Doppler peak (corresponding
to an angular size of
). Other peaks occur at harmonics of this. As the amplitude and
position of the primary and secondary peaks are intrinsically
determined by the number of electron scatterers and by the
geometry of the Universe, they can be used as a test of the
density parameter of baryons and dark matter, as well as other
cosmological constants.
Prior to the last scattering surface, the photons and matter interact on scales smaller than the horizon size. Through diffusion, the photons will travel from high density regions to low density regions `dragging' the electrons with them via Compton interaction. The electrons are coupled to the protons through Coulomb interactions, and so the matter will move from high density regions to low density regions. This diffusion has the effect of damping out the fluctuations and is more marked as the size of the fluctuation decreases. Therefore, we expect the Doppler peaks to vanish at very small angular scales. This effect is known as Silk damping [90].
Another possible diffusion process is free streaming. It occurs when collisionless particles (e.g. neutrinos) move from high density to low density regions. If these particles have a small mass, then free streaming causes a damping of the fluctuations. The exact scale this occurs on depends on the mass and velocity of the particles involved. Slow moving particles will have little effect on the spectrum of fluctuations as Silk damping already wipes out the fluctuations on these scales, but fast moving, heavy particles (e.g. a neutrino with 30 eV mass), can wipe out fluctuations on larger scales corresponding to 20 Mpc today [48].
Putting this all together, we see that on large angular scales
() we expect the CMB power spectrum to reflect the initially near
scale-invariant spectrum coming out of inflation; on intermediate
angular scales we expect to see a series of peaks, and on smaller
angular scales (< 10 arcmin) we expect to see a sharp decline
in amplitude. These expectations are borne out in the actual
calculated form of the CMB power spectrum in what is currently
the `standard model' for cosmology, namely inflation together
with cold dark matter (CDM). The spectrum for this, assuming
and standard values for other parameters, is shown in
Figure
2
.
The quantities plotted are
, versus
where
is defined via
and the
are standard spherical harmonics. The reason for plotting
is that it approximately equals the power per unit logarithmic
interval in
. Increasing
corresponds to decreasing angular scale
, with a rough relationship between the two of
radians. In terms of the diameter of corresponding proto-objects
imprinted in the CMB, a rich cluster of galaxies corresponds to a
scale of about 8 arcmin, while the angular scale corresponding to
the largest scale of clustering we know about in the Universe
today corresponds to 1/2 to 1 degree. The first large peak in the
power spectrum, at
's near 200, and therefore angular scales near
, is known as the `Doppler', or `Sakharov', or `acoustic'
peak.
As stated above, the inflationary CMB power spectrum plotted
in Figure
2
is that predicted by assuming the standard values of the
cosmological parameters for a CDM model of the Universe. In order
for an experimental measurement of the angular power spectrum to
be able to place constraints on these parameters, we must
consider how the shape of the predicted power spectrum varies in
response to changes in these parameters. In general, the detailed
changes due to varying several parameters at once can be quite
complicated. However, if we restrict our attention to the
parameters
,
and
, the fractional baryon density, then the situation becomes
simpler.
Perhaps most straightforward is the information contained in
the position of the first Doppler peak, and of the smaller
secondary peaks, since this is determined almost exclusively by
the value of the total
, and varies as
. (This behaviour is determined as mentioned above by the linear
size of the causal horizon at recombination, and the usual
formula for angular diameter distance.) This means that if we
were able to determine the position (in a left/right sense) of
this peak, and we were confident in the underlying model
assumptions, then we could read off the value of the total
density of the Universe. (In the case where the cosmological
constant was non-zero, we would effectively be reading off the
combination
.) This would be a determination of
free of all the usual problems encountered in local
determinations using velocity fields etc.
Similar remarks apply to the Hubble constant. The
height
of the Doppler peak is controlled by a combination of
and the density of the Universe in baryons,
. We have a constraint on the combination
from nucleosynthesis, and thus using this constraint and the
peak height we can determine
within a band compatible with both nucleosynthesis and the CMB.
Alternatively, if we have the power spectrum available to good
accuracy covering the secondary peaks as well, then it is
possible to read off the values of
,
and
independently, without having to bring in the nucleosynthesis
information. The overall point here is that the power spectrum of
the CMB contains a wealth of physical information, and that once
we have it to good accuracy and have become confident that an
underlying model (such as inflation and CDM) is correct, then we
can use the spectrum to obtain the values of parameters in the
model, potentially to high accuracy. This will be discussed
further below, both in the context of the current CMB data, and
in the context of what we can expect in the future.
![]() |
The Cosmic Microwave Background
Aled W. Jones and Anthony N. Lasenby http://www.livingreviews.org/lrr-1998-11 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |