# I Nonlinear integration

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1. Feb 9, 2017

### shinobi20

From cosmology, the friedmann equations are given by,
$H^2 = (\frac{\dot a}{a})^2 = \frac{8\pi G}{3} \rho \, , \quad \frac{\ddot a}{a} = -\frac{4\pi G}{3}(\rho+3p) \, , \quad$ where $\rho = \frac{1}{2}(\dot \phi^2 + \phi^2)$ and $p = \frac{1}{2}(\dot \phi^2 - \phi^2)$

To get $\dot H$,
$\dot H = \frac{d}{dt}(\frac{\dot a}{a}) = \frac{\ddot a}{a} - (\frac{\dot a}{a})^2 = -4\pi G(\rho + p) = -4\pi G \dot \phi^2$.

I want to solve for $H$ using this equation, where $0<t<10^7$. How should I solve this DE? It's ok if the solution is in the implicit form.

2. Feb 16, 2017

### Greg Bernhardt

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Feb 16, 2017

### Buzz Bloom

Hi shinobi:

When solving for H(t) I would generally work with a different form of the Friedmann equation.
The one I mean is in the article just above "Useful Solutions".

H is given as a function of a, together with the parameter H0, and several density ratio parameters, the Ωs with various subscripts.
Since H = (1/a) (da/dt), dt can be expressed in the form f(a) da. This can be numerically integrated to get t(a) for a specific value of a. I found the following online tool useful for this.
You many want to substitute a = e-x if you find problems with the tool when integrating the f(a) form.