# General Friedmann equation - how to solve this differential equation?

• beertje
beertje
Homework Statement
Solve the differential equation
Relevant Equations
(adot / a)^2 = k / a^q

Hello fellow physicists, I am taking a course "Introduction to Cosmology" and I am asked to solve this equation called the Friedmann equation. I understand what it represents (scale factor of cosmic time) but I have no idea how to solve this differential equation, even though I took a whole course on solving those (Euler-Lagrange etc.)

Please give me a small pointer :)

Take the square root (which sign is the right one?) and then separate variables.

beertje
vanhees71 said:
Take the square root (which sign is the right one?) and then separate variables.
Thanks! That was way more obvious than I thought. Summer break always breaks me up.

PhDeezNutz and vanhees71

## What is the General Friedmann Equation?

The General Friedmann Equation is a set of equations derived from Einstein's field equations of General Relativity. They describe the expansion of the universe in the context of a homogeneous and isotropic cosmological model. The most common form includes terms for the Hubble parameter, the density of different components (matter, radiation, dark energy), and the curvature of the universe.

## What are the variables and parameters in the Friedmann Equation?

The primary variables and parameters in the Friedmann Equation include the scale factor $$a(t)$$, which describes how distances in the universe expand or contract over time; the Hubble parameter $$H(t)$$, which is the rate of expansion of the universe; the density parameters for matter $$\Omega_m$$, radiation $$\Omega_r$$, and dark energy $$\Omega_\Lambda$$; and the curvature parameter $$k$$, which can be $$-1$$, $$0$$, or $$1$$ corresponding to open, flat, or closed universes, respectively.

## How do you solve the Friedmann Equation for a flat universe with only matter and dark energy?

For a flat universe ($$k = 0$$) with only matter and dark energy, the Friedmann Equation simplifies to $$H^2 = H_0^2 \left( \Omega_m a^{-3} + \Omega_\Lambda \right)$$. To solve this differential equation, you can separate variables and integrate. This typically involves expressing $$H$$ as $$\dot{a}/a$$ and integrating with respect to $$a$$ to find $$a(t)$$, the scale factor as a function of time.

## What initial conditions are needed to solve the Friedmann Equation?

To solve the Friedmann Equation, you need initial conditions such as the value of the scale factor $$a(t)$$ at a particular time $$t$$. Commonly, the present time $$t_0$$ is chosen with $$a(t_0) = 1$$. Additionally, the current values of the Hubble parameter $$H_0$$ and the density parameters $$\Omega_m$$, $$\Omega_r$$, and $$\Omega_\Lambda$$ are required.

## Are there analytical solutions to the Friedmann Equation?

Analytical solutions to the Friedmann Equation exist for some specific cases, such as a universe dominated by a single component (e.g., matter-only or radiation-only). For more complex scenarios involving multiple components (matter, radiation, dark energy) or non-zero curvature, numerical methods are typically employed to solve the equation. These involve discretizing the equation and using computational algorithms to approximate the solution.

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