Is the universe open or closed?
where DL is the luminosity distance.
It is useful to have an estimate of the relationship between the Hubble time, H-1, and the true age of the universe, t0. Obviously, this depends on which model we use to describe the universe.
For the Friedmann models (A = 0) the results are as follows.
For a flat universe (k = 0), t0 = (2/3) H-1 (20.40a)
For negative curvature (k = -1)
It is also useful to have an estimate of the relationship between the age of the universe when a photon was emitted and the redshift. In one model it is given by t (Gyr) = 10.5 (H0/65 km/s/Mpc) (1 + z)-3/2 (20.41)
In this section we look at evidence that might allow us to decide whether the universe is open or closed. It is impressive that we can even ask such a question and hope to achieve an answer.
The basic question is whether the actual density is less than or greater than the critical density. We could start by adding up the density of all the matter we can see, to find out if it gives P > pcrit. However, we already know that there is a problem with dark matter, so, if we only include the visible matter, we will be missing a significant amount. Of course, if the visible matter is sufficient to close the universe, then we don't have to worry about the dark matter. It turns out that if the visible matter is insufficient to close the universe, then we have to account for the dark matter. It turns out that the density of visible matter is about 1% of the critical density.
If the universe is to be closed, the dark matter must do it. From Table 20.2, we see that the amount of dark matter required to close the universe is greater than the dark matter in clusters of galaxies. There also appears to be a trend towards more dark matter on larger scales. Therefore, we would not be surprised if there is enough dark matter to close the universe. However, in our attempt to see if the universe is open or closed, we can only include dark matter that we know is present by its gravitational effects on visible matter. We can therefore include the dark matter in galaxies, and clusters of galaxies, since we can detect its gravitational
Table 20.2. I Mass-to-light ratios on different scales.
MIL (solar units)
Milky Way to Sun Spiral galaxy disk Elliptical galaxy Halo of giant elliptical Rich cluster of galaxies To close the universe
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