Age of the Universe: Intuitive Understanding

[Ichimaru]

I've been doing some Cosmology, but I'm having a really hard time understanding the results for the age of the Universe intuitively. For example I can work out from the FRW equation that in the case of no Cosmological constant and no radiation in a flat matter dominated universe the age is approximately:

\begin{equation}

t_{0} = \frac{2}{3H_{0}}

\end{equation}

And in the case of no cosmological constant, no radiation, no matter and only curvature:


\begin{equation}

t_{0} = \frac{1}{H_{0}}

\end{equation}

However I don't understand the results physically. Are there any good ways about thinking about these parameters and their effects on the age of the Universe today?

 

[Simon Bridge]

By "parameters" you mean the "cosmological constant", "radiation", and "matter", parameters?

You know you can get an estimate of the age of the Universe from it's size and knowing that it is expanding.
From there it is a matter of making some sort of assumptions about how the expansion went in the past.

The FRW equation (etc) tells you how each parameter affects the calculated age. That's probably the best way to understand them.

Perhaps if you talk about how you currently understand the different parameters, we will be able to see where you need help?

 

[George Jones]

And in the case of no cosmological constant, no radiation, no matter and only curvature:\begin{equation}

t_{0} = \frac{1}{H_{0}}

\end{equation}

This case is just special relativity, so my guess is that this is something like distance = rate times time, but, because the coordinates used aren't standard inertial coordinates, the interpretation might be a bit subtle.

Why special relativity when in the case when there is "only curvature"? Because in this context, "curvature" refers to spatial curvature, not spacetime curvature. Spacetime curvature is zero in this case.

You're talking about the Milne universe, which is a a patch of Minkowski spacetime in somewhat unusual coordinates.

Start with Minkowski spacetime in spherical coordinates,

$$ds^2 = dt'^2 - dr^2 - r^2 d \Omega^2 ,$$

and make the coordinate transformation

$$t' = t \cosh \chi$$

$$r = t \sinh \chi.$$