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.

 Figure 1. An example of a power spectrum deduced from a large-scale structure (LSS) survey. A megaparsec (Mpc) is 3.26 x 1024 cm, and h is the dimensionless Hubble constant, H0 = 100h km s-1 Mpc-1. A wavenumber k is roughly related to a length scale of 2 / k.