**2.1 The Power Spectrum From Large-Scale Structure**

The power spectrum is related to the *rms* fluctuations in the density
on scale *R* = 2 */ k*. The
exact relationship depends upon sampling
procedure, window functions, etc. But for a simple intuitive feel,
imagine we have mass points spread throughout some sample volume. Now
place a sphere of radius *R* in the volume and count the number of
points within the sphere. Then repeat as often as you have the time
or patience to do so. There will be an average number <*N*>, and
an *rms* fluctuation <(
*N* / <*N*> )^{2} >^{1/2}. The power
spectrum is related to that *rms* fluctuation: (*k* = 2 *R*^{-1})
<( *N* / <*N*> )^{2}
>^{1/2}. Repeating the procedure
for many values of *R* will give as a function of *R*
- the power spectrum.

Now how does one go about observing the mass within a sphere of radius
*R*? Well, it is difficult to measure the mass. It is easier to
count the number of galaxies. So one assumes that the galaxy
distribution traces the mass distribution. Although it seems
reasonable that regions of high density of galaxies correspond to
regions of high mass density, since most of the mass is dark, the
proportionality might not be exact. Thus, we have to allow for a
possible *bias* in the power spectrum. Other problems also arise.
The distance to an object is not measured directly; what is measured
is its redshift. The redshift is determined by the distance to the
object, as well as its peculiar motion. In regions of large
overdensity the peculiar motions may be large, resulting in what is
known as redshift distortions. Another problem is nonlinear
evolution, which distorts the power spectrum in regions of large
overdensity. Thus, if one wants to compare the observed power
spectrum with the linear power spectrum generated by early-universe
physics, it is necessary to make corrections for bias, redshift-space
distortions, and nonlinear evolution.

Deducing the power spectrum from galaxy surveys is a tricky business.
Rather than go into the details, uncertainties, and all that, I will
just present a representative power spectrum in
Fig. (1).
Note that *(k)* decreases
with increasing length scale
(decreasing wavenumber). The universe is lumpy on small scales, but
becomes progressively smoother when examined on larger scales.