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I'm studying the nonrelativistic-matter perturbations if the expansion of the Universe is driven by a combination of components.

I'm currently Following this document (The growth of density perturbations) from Caltech. However, the author doesn't explain how he has found the solutions for the following expression that can be found in page 5.

$$\delta'' + \frac{2+3y}{2y(1+y)}\delta' - \frac{3}{2y(1+y)}\delta = 0$$

$$y=\frac{a}{a_{eq}}$$

Moreover, I found those notes (University of Cambridge Part II Mathematical Tripos) from David Tong at Cambridge explaining the transfer function (p.146).

Thus, I'm wondering if there is a relationship between the initial conditions and the transfer function. However, the first document doesn't mention the initial conditions for the solutions.

My thoughts are that we can find 2 independents solutions, one for small wavelengths and one for long wavelength as explained by Tong. Thus, $T(k) \approx C$ for long wavelengths and $T(K) \approx C ln(a?)$ for short wavelengths.

Furthermore, if the transfer function is defined as follow. $$\delta(\vec{k},t_0) = T(k)\delta(\vec{k},t_i)$$ Does it means that $T(k)$ acts as the constant in the particular solution?

I might be totally wrong. There is a lot of guesses here, since I'm unsure to understand.

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  • $\begingroup$ Minor comment to the post (v1): Please consider to mention explicitly author, title, etc. of links, so it is possible to reconstruct links in case of link rot. $\endgroup$
    – Qmechanic
    Commented Apr 7 at 11:36

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Deep in radiation domination ($y\ll 1$), the two solutions are $\delta=1$ and $\delta=\ln a$. Physically, these solutions describe the effects of particle drift in the complete absence of peculiar forces, which is appropriate because the dominant radiation is spatially uniform, so it does not exert peculiar forces.

However, peculiar forces were not always absent. There are radiation perturbations at scales larger than the cosmological horizon ($k\ll aH$). At horizon entry ($k\simeq aH$), those perturbations briefly source significant gravitational potentials, which give the matter particles an initial "kick". The details of this process are what sets the particular combination of $\delta=1$ and $\delta=\ln a$. Hu & Sugiyama (1996) is the classic reference for this and can tell you what the combination is for each $k$ (see especially Appendix B).

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