# The Friedmann equation in a lambda-dominated universe

• _Andreas
In summary, the conversation discusses the relationship between a lambda-dominated euclidean universe and exponential expansion. It also explores the Friedmann equation and its different representations, with a conclusion that using both a constant term lambda and a vacuum energy source term is redundant.

## Homework Statement

Show that a lambda-dominated euclidean universe entails an exponential expansion.

## Homework Equations

Equation of state is P=-E (E=epsilon)
There is no curvature (i.e. a flat universe)

The Friedmann equation is then

((a(dot)/a)^2)=((8piG)/3c^2)E + lambda/3 (eq. 1)

which -- and this is where I get lost -- is equivalent to

((a(dot)/a)^2)=((8piG)/3c^2)Elambda (eq. 2)

## The Attempt at a Solution

lambda/3 = ((8piG)/3c^2)Elambda, but if I make this substitution into the Friedmann equation, I get

((a(dot)/a)^2)=((8piG)/3c^2)E + ((8piG)/3c^2)Elambda

What am I doing wrong? I'm after (eq. 2).

You have two different representations of the same thing in the Friedmann equation. A constant term lambda is the same thing as a vacuum energy source term satisfying rho=-p. Which one do you want to work with? You don't need both.

I'm not sure I understand. Could you elaborate further?

_Andreas said:
I'm not sure I understand. Could you elaborate further?

What part don't you understand? A fluid with E=-p satisfies dE/dt=0 by the conservation equation, so is a constant. lambda is also a constant. You have two constants. It's a little redundant.

I just wanted to say thank you for your efforts, Dick. They're appreciated. I forgot to say that. Sorry!

## 1. What is the Friedmann equation?

The Friedmann equation is a mathematical expression that describes the evolution of the universe in the context of general relativity. It relates the rate of expansion of the universe (Hubble parameter) to the density of matter and energy in the universe.

## 2. What does the "lambda-dominated" universe refer to?

The "lambda-dominated" universe refers to a universe where the dominant form of energy is the cosmological constant (lambda), which is thought to be responsible for the accelerated expansion of the universe.

## 3. How does the Friedmann equation change in a lambda-dominated universe?

In a lambda-dominated universe, the Friedmann equation takes on a slightly different form, as it includes the contribution of the cosmological constant to the overall energy density of the universe. This leads to a faster expansion of the universe compared to a universe dominated by matter.

## 4. What are the implications of the Friedmann equation in a lambda-dominated universe?

The Friedmann equation in a lambda-dominated universe has several implications, including the prediction of an ever-expanding universe and the existence of dark energy (the cosmological constant) as a driving force behind this expansion. It also helps to explain the observed large-scale structure of the universe.

## 5. How does the Friedmann equation relate to the Big Bang theory?

The Friedmann equation is a key component of the Big Bang theory, as it is used to describe the expansion of the universe from a singularity at the beginning of time. It helps to explain how the universe has evolved over time and provides a framework for understanding the origin and structure of the universe.