Starring: Doug McClure, spatial curvature
Spatial curvature was completely irrelevant at z>1000, when
the acoustic oscillations were created. Omega_k therefore
doesn't change the shape of the peaks at all - it merely
shifts them sideways, since the conversion from the physical
scale of the wiggles (in meters) into the angular scale
(in degrees, or multipole l) depends on whether space is curved.
If space has negative curvature (positive Omega_k), like
a Pringles potato chip or a saddle, then the angle subtended on the
sky decreases, shifting the peaks to the right.
If space has positive curvature, like a balloon, then the
peaks shift to the left.
Meanwhile, you'll see that the galaxy power spectrum shifts both
horizontally (in the opposite direction) and vertically.
Let's start with the vertical part. Curvature slows the growth
of fluctuations and therefore lowers the curve.
The horizontal shift wouldn't occur if the horizontal axis had
physical distance units on it (meters or Mpc, say). But it uses
astronomical distance units (Mpc/h), so changing the Hubble
parameter h shifts the curve.
Did somebody say h? Weren't we just changing the curvature?
Of the 12 parameters shown,
only 11 are independent.
So when we change the curvature with
Lambda and all the matter densities fixed, then
the Hubble constant automatically changes too.
Finally,if you've patiently read this far, let's return to the CMB
In addition to shifting the peaks sideways, curvature also
causes fluctuations in the gravitational field to decrease over time.
This means that if a photon flies though a potential well on the way
to us, its blueshift from falling in will exceed
its redshift from climbing out. Since these extra fluctuations
in the photon temperature (known as the late ISW effect)
happen at late times, they show up
on large scales (to the left in the power spectrum).