- #1

chirag1

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- TL;DR Summary
- How to find the cutoff variable to halt the integration of the cosmological perturbations top determine the evolution of different k-modes?

I'm trying to numerically evolve the Einstein-Boltzmann equations for cold dark matter perturbations using Runge-Kutta method of the fourth order. There are 5 standard equations:

$$

\begin{align}

\dot{\Theta}_{r,0}+k\Theta_{r,1}&=-\dot{\Phi} \\

\dot{\Theta}_{r,1}+\frac{k}{3}\Theta_{r,0} & =\frac{-k}{3}\Phi \\

\dot{\delta}+ikv &= -3\dot{\Phi} \\

\dot{v}+\frac{\dot{a}}{a}v &= ik\Phi \\

\dot{\Phi}&=\frac{1}{3\dot{a}}\frac{3H_{0}^{2}}{2}\left(\Omega_{m}\delta+4\Omega_{r}\Theta_{r,0}a^{-1}\right)-ak^{2}\Phi-\frac{\dot{a}}{a}\Phi

\end{align}

$$The problem is, we cannot integrate them all the way to the present as radiation moments are difficult to track at late times and especially so for small scale (large k) modes. The solution to this is to find a cutoff time at which we halt the integration, discard the radiation perturbations and restart the integration. I'm facing the issue of how to obtain an expression for this cutoff time here, which depends on the k-mode. I'm more surprised by the lack of presented solutions for this standard problem (this numerical integration task is given as an textbook exercise in Chapter 8 (ex. 8.2) Modern Cosmology-Dodelson 2nd edition and 1st edition also which was more than 15 years ago, but there is no solution to this textbook exercise as well!) in literature or papers.

I've tried a lot to find something but everyone is seemingly not tackling these 5 equations and taking a different approach. But for my project, I've to work on these 5 equations only. The closest I got to something was Florian Borchers' thesis: https://www.imperial.ac.uk/media/im...ations/2010/Florian-Borchers-Dissertation.pdf where they give an expression for cutoff conformal time (page 32) but give no explanation. That expression is:

$$

\eta_{\text{stop}} = \eta_{\text{today}} - \frac{2}{3}log(100k/h)

$$

They actually use stepperdopr853 method for integration and conformal time as their integration variable, while I use RK4 and scale factor. I've tried to account for it and take help of chatgpt as well and all literature that I could find but in vain. I'm very stuck.

$$

\begin{align}

\dot{\Theta}_{r,0}+k\Theta_{r,1}&=-\dot{\Phi} \\

\dot{\Theta}_{r,1}+\frac{k}{3}\Theta_{r,0} & =\frac{-k}{3}\Phi \\

\dot{\delta}+ikv &= -3\dot{\Phi} \\

\dot{v}+\frac{\dot{a}}{a}v &= ik\Phi \\

\dot{\Phi}&=\frac{1}{3\dot{a}}\frac{3H_{0}^{2}}{2}\left(\Omega_{m}\delta+4\Omega_{r}\Theta_{r,0}a^{-1}\right)-ak^{2}\Phi-\frac{\dot{a}}{a}\Phi

\end{align}

$$The problem is, we cannot integrate them all the way to the present as radiation moments are difficult to track at late times and especially so for small scale (large k) modes. The solution to this is to find a cutoff time at which we halt the integration, discard the radiation perturbations and restart the integration. I'm facing the issue of how to obtain an expression for this cutoff time here, which depends on the k-mode. I'm more surprised by the lack of presented solutions for this standard problem (this numerical integration task is given as an textbook exercise in Chapter 8 (ex. 8.2) Modern Cosmology-Dodelson 2nd edition and 1st edition also which was more than 15 years ago, but there is no solution to this textbook exercise as well!) in literature or papers.

I've tried a lot to find something but everyone is seemingly not tackling these 5 equations and taking a different approach. But for my project, I've to work on these 5 equations only. The closest I got to something was Florian Borchers' thesis: https://www.imperial.ac.uk/media/im...ations/2010/Florian-Borchers-Dissertation.pdf where they give an expression for cutoff conformal time (page 32) but give no explanation. That expression is:

$$

\eta_{\text{stop}} = \eta_{\text{today}} - \frac{2}{3}log(100k/h)

$$

They actually use stepperdopr853 method for integration and conformal time as their integration variable, while I use RK4 and scale factor. I've tried to account for it and take help of chatgpt as well and all literature that I could find but in vain. I'm very stuck.