General Friedmann equation - how to solve this differential equation?

  • Thread starter beertje
  • Start date
In summary, the Friedmann equation is a key differential equation in cosmology that describes the expansion of the universe. To solve it, one typically employs techniques from differential equations, such as separation of variables or numerical methods, depending on the complexity of the matter-energy content and curvature of the universe. Solutions often involve integrating the equation to find relationships between scale factor, time, and cosmic density parameters, leading to insights into the universe's evolution and fate.
  • #1
beertje
3
2
Homework Statement
Solve the differential equation
Relevant Equations
(adot / a)^2 = k / a^q
friedman.png


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 :)
 
Physics news on Phys.org
  • #2
Take the square root (which sign is the right one?) and then separate variables.
 
  • Like
Likes beertje
  • #3
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.
 
  • Haha
  • Like
Likes PhDeezNutz and vanhees71

FAQ: General Friedmann equation - how to solve this differential equation?

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.

Similar threads

How Do You Solve These Commutator Relations?
Replies
2
Views
1K
Solve these two coupled first-order differential equations and sketch the flow
Replies
2
Views
1K
Lagrangian for the electromagnetic field coupled to a scalar field
Replies
7
Views
3K
Current through Ballistic 2DEG Channel
Replies
1
Views
1K
Investigating Lagrangians and Constraints for Tension Calculation
Replies
5
Views
1K
Can This Differential Equation Be Solved by Separation of Variables?
Replies
7
Views
3K
State space controllers - find equations from differentials
Replies
12
Views
1K
Rewriting Friedmann Eq. with Conformal Time & Density Params
Replies
4
Views
3K
First Friedmann Equation forms
Replies
1
Views
2K
GR:KvFs & Geodesics: Solving for L & Derivatives
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top