# Interpretting the cosmological constant

1. Jun 5, 2015

### bjb406

I am trying to wrap my brain around the idea of the cosmological constant, and how it gives the result of an accelerating universe.

I have read different explanations for it, and in some it gives me the impression that it is an actual physical force, pushing everything away from everything else. While in others I get the impression I should look at it as thought space itself is expanding, meaning that even though 2 objects aren't actually being pushed away from each other, the space between them is expanding.

So which one is it? Or am I totally misunderstanding the whole thing?

2. Jun 5, 2015

### Chalnoth

The super short version is this:

The amount of space-time curvature is given by the amount of stuff in the universe. If that stuff doesn't change over time, then the space-time curvature is a constant.

The space-time curvature itself manifests as the expansion of our universe. So a constant amount of stuff means a constant rate of expansion. The rate of expansion is defined as:

$$H(t) = {1 \over a}{da \over dt}$$

Here $H$ is the Hubble parameter and $a$ is the scale factor (when the scale factor doubles in value, things are twice as far apart). Now, if $H(t)$ is a constant (we'll call it $H_0$), then we have the differential equation given by:

$${da \over dt} = H_0 a$$

If you know your differential equations at all, you should know that this is the equation for exponential growth. The solution being:

$$a(t) = a(t=0) e^{H_0 t}$$

This may make intuitive sense if you realize that the Hubble parameter is the fractional increase in distance per unit time. So if $H_0 = 0.01/s$, then distances will increase by 1% per second. So after one second, a distance of 1 meter becomes a distance of 1.01 meters. After 2 seconds it's a distance of 1.02 meters. After 10 seconds, it's 1.105 meters. After a minute, it's 1.822 meters. After ten minutes, it's 403.4 meters. As you can see, the rate at which this distance increases over time increases dramatically. At each second, the rate is 1% of the current distance. So as the current distance gets larger, the increase in the distance gets larger as well.

I should mention that it's also possible to write down a force by translating the General Relativity solution into Newtonian terms. In that situation, the acceleration due to gravity picks up an extra term proportional to the cosmological constant times the distance between any two objects.