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

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• shinobi20
In summary, the Friedmann equations in cosmology relate the expansion of the universe (H) to the density (ρ) and pressure (p) of the matter in the universe. To solve for the time derivative of H (H'), one can use an alternate form of the Friedmann equation and numerically integrate it using a tool like "SolveMyMath." It may be helpful to substitute a different variable (such as e^-x) when using the tool.
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.

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.

Regards,
Buzz

## 1. What are the Friedmann equations?

The Friedmann equations are a set of equations that describe the expansion of the universe in the context of general relativity. They were derived by the Russian physicist Alexander Friedmann in the 1920s and are used to model the evolution of the universe over time.

## 2. What is a nonlinear differential equation?

A nonlinear differential equation is a mathematical equation that involves a function and its derivatives, where the function is raised to a power or contains other nonlinear terms. These equations are more complex and difficult to solve compared to linear differential equations, which only include linear terms.

## 3. Why is it important to solve nonlinear differential equations in the context of cosmology?

Nonlinear differential equations are essential for understanding the complex dynamics of the expanding universe. The Friedmann equations, which are nonlinear, allow us to model the behavior of the universe and make predictions about its past, present, and future.

## 4. What is the value of Hubble's constant (H0) in the Friedmann equations?

Hubble's constant, denoted as H0, represents the current rate of expansion of the universe. Its value is estimated to be approximately 67.4 kilometers per second per megaparsec, meaning that for every megaparsec (a unit of distance used in astronomy), the universe expands by 67.4 kilometers per second.

## 5. How do we solve the Friedmann equations for H0-107?

To solve the Friedmann equations for H0-107, we must first plug in the appropriate values for the other variables in the equation, such as the matter density, dark energy density, and radiation density. Then, we can use numerical methods or approximations to find the value of H0-107 that satisfies the equation.

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