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The universe is not exactly homogeneous and isotropic, but it is a sufficiently accurate description of the universe on large scales that it is useful to consider homogeneity and isotropy as a first approximation, and discuss departures from this idealized smooth universe.

Let us begin by considering the density field, rho (vectorx). If the average density of the universe is denoted as < rho >, then we can define a dimensionless density contrast delta (vectorx) as

Equation 1 (1)

Of course we cannot predict delta (vectorx), but we can hope to predict the statistical properties of delta (vectorx). The correct arena to discuss the statistical properties of the density field is in Fourier space, where one decomposes the density contrast into its various Fourier modes delta(vectork):

Equation 2 (2)

where V is some irrelevant normalization volume. After a little Fourier manipulation and some mild assumptions about the density field, it is easy to show that the two-point autocorrelation function of the density field can be expressed solely in terms of | delta (vectorx) |2:

Equation 3 (3)

where A is yet another irrelevant constant.

So long as the fluctuations are Gaussian, all statistical information is contained in a quantity known as the power spectrum, which can be defined as either

Equation 4 (4)

The first choice is much more physical, as it represents the power per logarithmic decade in the fluctuations. Although the first choice makes much more sense, the second choice is what is usually used. It turns out that graphs of P (k) have a nicer form (but less physical content) than corresponding graphs of Delta2 (k). Since the widespread availability of color graphics, presentation seems to be everything, and information content of secondary concern.

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