Why does non-baryonic matter give structure formation a head start? This image is from the textbook by Ryden. It shows that the density perturbation was able to grow earlier if there were non-baryonic dark matter, why is that?

 A: This is because non-baryonic dark matter does not interact with radiation, hence it can collapse and form potential wells before baryonic matter can (and the potential wells attract baryonic matter).
See Wiki (most relevant parts highlighted):

Structure formation refers to the period after the Big Bang when density perturbations collapsed to form stars, galaxies, and clusters. Prior to structure formation, the Friedmann solutions to general relativity describe a homogeneous universe. Later, small anisotropies gradually grew and condensed the homogeneous universe into stars, galaxies and larger structures. Ordinary matter is affected by radiation, which is the dominant element of the universe at very early times. As a result, its density perturbations are washed out and unable to condense into structure. If there were only ordinary matter in the universe, there would not have been enough time for density perturbations to grow into the galaxies and clusters currently seen.
Dark matter provides a solution to this problem because it is unaffected by radiation. Therefore, its density perturbations can grow first. The resulting gravitational potential acts as an attractive potential well for ordinary matter collapsing later, speeding up the structure formation process.

