Physical scale of the
peaks corresponds to the distance sound
Angular scale of the
peaks measured in experiments
Relationship between the two gives the spatial
curvature of the universe
A flat universe has the so-called critical
With the beautiful agreement between the predications
and observations of the shape of the first peak, we can now draw inferences
from its angular position at l~200 based on the interpretation that
we are seeing the first compression of a sound
wave before recombination.
The key is that we know the physical
scale associated with the peak: the distance
sound can travel before recombination, the so-called sound
horizon. We measure the angular
scale from the experiments themselves.
Combining the two tells us the spatial curvature
of the universe. Cosmologists call this an angular
diameter distance test for curvature.
Let's see how that works. Think of the more
familiar curvautre of the surface of the Earth and imagine you are the
observer at the pole. As you look out at a fixed angle, you are looking
along lines of constant longitude.
Since the surface curves, the distance between the two lines is smaller
than it would be if the Earth were flat:
In a closed universe, when you look out at a fixed
angular scale you are therefore looking at a smaller physical scale.
Conversely, for a fixed physical scale, in our case the distance sound
can travel, you have to look out at a larger angle (or lower multipole
l) to see its full extent.
The spatial curvature of the universe is related
to the total energy density of
all consituents of the universe. You can alternately think of what
is doing the curving of the paths of the CMB photons from recombination
as gravitational lensing from the intervening energy density.
The upshot is that if the universe is flat,
it has a very special energy density called the critical
The alert reader will note that this
argument shoves one thing under the rug. The usual use of comparing
physical to angular size is to measure the distance
to an object. Think about how you might
judge the distance to an on coming car at night. You look at how
far in angle the headlights are separated and knowing how wide a car typically
is, you intuitively infer its distance. You would be thought strange
if hypothesized a giant magnifying lens in between making the headlight
separation look larger! So how is that we measure mainly the curvature
of the universe and not the distance to recombination? Fortunately
for us, the ambiguities in the distance scale
out since they change the distance sound can
travel before recombination and the distance CMB light can travel after
recombination in the same way. That's just like redefining
what you mean by a meter and doesn't affect angular scales at all.
The residual effects are small as we shall see later but are currently
a subject of great interest.
Finally, you should be aware that there is a difference
between spatial curvature and curvature of
space-time. Even if space is flat, that
the universe is expanding at all is evidence that Einstein was right about
space-time being curved!