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hidenori
hidenori
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I'm trying to learn to numerically solve the perturbed Boltzmann-Einstein equations in cosmology using the RK4 method. These are the equations:

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

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

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

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

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

How do I go about this? I am feeling lost to even start.

I'm trying to learn to numerically solve the perturbed Boltzmann-Einstein equations in cosmology using the RK4 method. These are the equations:

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

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

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

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

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

How do I go about this? I am feeling lost to even start.

I'm trying to learn to numerically solve the perturbed Boltzmann-Einstein equations in cosmology using the RK4 method. These are the equations:

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

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

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

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

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

How do I go about this?

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Anton Menshov ♦
Anton Menshov ♦
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I'm trying to learn to numerically solve the perturbed Boltzmann-Einstein equations in cosmology using the RK4 method. These are the equations:

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

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

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

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

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

How do I go about this? I am feeling lost to even start.

I'm trying to learn to numerically solve the perturbed Boltzmann-Einstein equations in cosmology using the RK4 method. These are the equations:

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

$\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}(\Omega_{m}\delta+4\Omega_{r}\Theta_{r,0}a^{-1})-ak^{2}\Phi-\frac{\dot{a}}{a}\Phi $

How do I go about this? I am feeling lost to even start.

I'm trying to learn to numerically solve the perturbed Boltzmann-Einstein equations in cosmology using the RK4 method. These are the equations:

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

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

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

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

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

How do I go about this? I am feeling lost to even start.

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hidenori
hidenori
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Solving perturbed Einstein Boltzmann equations using RK4

I'm trying to learn to numerically solve the perturbed Boltzmann-Einstein equations in cosmology using the RK4 method. These are the equations:

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

$\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}(\Omega_{m}\delta+4\Omega_{r}\Theta_{r,0}a^{-1})-ak^{2}\Phi-\frac{\dot{a}}{a}\Phi $

How do I go about this? I am feeling lost to even start.