What size of object does the peak of the cosmological power spectrum correspond to?

Question

The title almost says it all, but to flesh it out more, what is the size a sphere corresponding to the peak in the cosmological power spectrum (Figure 2: https://ned.ipac.caltech.edu/level5/Sept11/Norman/Norman2.html).

It would be great to get a feel of both the size of the collapsing region (e.g. the first stars collapsing in clusters from a region with size of order kpc) and the size that such an object could theoretically collapse down to (if it could cool effectively and gravity wasn't swamped by other forces).

Answer 1

From the figure in the website you link, we see that the peak in the power spectrum occurs at a wavelength of about 300 [h$^{-1}$/Mpc]. The value of h is 0.68 for a Hubble constant of 68 km/s/Mpc (a value based on the Planck 2013 results). If we take 300 and divide by 0.68 to get in units of Mpc and round to the nearest ten we get

$$\lambda = 440 ~\textrm{Mpc}.$$

This is much larger than any galaxy or even galaxy cluster (galaxy clusters are $\sim 1-10$ Mpc in size). We have to get to structures known as superclusters, which are groups of clusters, or galaxy filaments, which are so named because they usually longer in one direction than the other two directions.

The Laniakea Supercluster, which the Milky Way is part of, is close to this size but still a little small, at 160 Mpc.

The Sloan Great Wall, a galaxy filament, is about 400 Mpc in length.

In short, the length scale that relates to the peak in the power spectrum of matter fluctuation densities encompasses the largest structures that have been discovered in the universe.