I was reading a paper on cosmological perturbation in modified $f(R)$ gravity link https://arxiv.org/abs/0802.2999. From eqn 34 in the paper, they plot $\delta$ vs $1+z$ in figure 1. How to set the initial condition for $\dot{\delta}$ and $\delta$?
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$\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$– Community BotCommented Feb 16 at 3:13
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The initial $\delta$ is totally arbitrary, as in first-order perturbation theory, the evolution is independent of the amplitude. This just leaves the matter of the initial $\dot\delta$.
For Newtonian gravity (and hence general relativity in the subhorizon limit), during the matter epoch, it is most natural to just set $\dot\delta=H\delta$, where $H$ is the Hubble rate. I believe this is also true for modified gravity theories such as that discussed in the paper you link, since they are designed to match general relativity at early times.
The reason for this choice is that the two solutions during matter domination are the growing mode $\delta\propto a$ and the decaying mode $\delta\propto a^{-3/2}$, where $a$ is the scale factor. The decaying mode quickly becomes negligible, so all that is left is the growing mode. But for the growing $\delta\propto a$ mode, $$\dot\delta=\frac{\mathrm{d}\delta}{\mathrm{d}a}\frac{\mathrm{d}a}{\mathrm{d}t} = H\delta.$$
Note if you pick an arbitrary initial $\dot\delta$, you will in general excite both growing and decaying modes, but your solution will rapidly converge to pure growing mode.
