# De Sitter universe, is it compatible with darkenergy?

1. Dec 19, 2009

### hurk4

Maybe obvious, but I will ask it here.
Is a "De Sitter Universe (as a model) compatible with dark energy, and if so, is it then also compatible with constant-dark-energy-density?

kind regards
Hurk4

2. Dec 19, 2009

### sylas

As I understand it, a De Sitter Universe is what you get when the universe is dominated by dark energy, with a constant energy density.

Cheers -- sylas

3. Dec 20, 2009

### hurk4

Dear Sylas.

As a (math. model):
1) also when this domination is exact 100% ?
2) is it static?
3) is its space-time restricted?

regards
hurk4

4. Dec 20, 2009

### sylas

I am not sure what you mean. I'm a novice in cosmology, so I'm not good with terminology, sorry. The simplest case, which is what de Sitter worked out, I believe, is 100% dark energy. In this case, the universe is static in an interesting kind of way. It is always expanding, but the Hubble constant is fixed for all time. There's also no singularity in the past, it simply remains at 100% dark energy forever. This is also a flat universe, and so (ignoring topological oddities) it would be infinite. Infinite in space, in time to the future, and in time to the past.

The scale factor is
$$a = e^{Ht}$$​
t is time, and H is the Hubble constant. The hubble constant at a point in time is defined to be
$$\frac{\dot{a}}{a} = \frac{He^{Ht}}{e^{Ht}} = H$$​

The differential equation for scale factor in the simple FRW models is
$$\dot{a} = a H_0 \sqrt{\Omega_r a^{-4} + \Omega_m a^{-3} + \Omega_k a^{-2} + \Omega_\Lambda}$$​

100% dark energy means ΩΛ is 1 and the other densities are zero, so it becomes the very simple equation
$$\dot{a} = a H_0$$​
from which we get the exponential function for the scale factor.

Cheers -- sylas

5. Dec 25, 2009

### hurk4

Static?, (in an interesting kind of way)
Why then t dependent?

Regards
hurk4

6. Dec 25, 2009

### Ich

It isn't, if you switch to static coordinates. It's all explained in the Wikipedia article.

7. Dec 26, 2009

### hurk4

Ok, thank you.

regards hurk4

8. Dec 31, 2009

### hurk4

Could or did De Sitter foresee in his time that his model was compatible with the acceleration? Can we see from his model how this acceleration works out, e.g. linear or exponential or?

kind regards
hurk4