Solve Nonlinear DE: Friedmann Eqns for H 0-10^7

[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.

 

[Buzz Bloom]

Hi shinobi:

When solving for H(t) I would generally work with a different form of the Friedmann equation.

https://en.wikipedia.org/wiki/Friedmann_equations​

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.

http://solvemymath.com/online_math_calculator/calculus/definite_integral/index.php .​

You many want to substitute a = e^-x if you find problems with the tool when integrating the f(a) form.

I hope this is helpful.

Regards,
Buzz