Estimate the age of the universe!

We can easily estimate the age of the universe if we assume that the universe has always been expanding at the current rate. Since the galaxies are currently flying away from each other, we can run the expansion back in time, and ask how long it would take for all the galaxies in the universe to come back together again.

First of all, remind yourself of the basic relationship between velocity, distance and time:

time = distance / velocity.

If we know the distance between M100 and the Milky Way, and their relative velocity, we can calculate how long it took them to travel their distance of separation.

age of universe = [ (distance to M100) / (recessional velocity of M100) ]

In other words,

age = d / v.

Recall that

H_o = v / d,

so the age of the universe is

age = 1 / H_o.

That seems simple enough. There's only one little hitch... We used units of (km/sec/Mpc) for H_o in the previous section. Inverting these gives units of (sec Mpc / km) for the age of the universe. How long is a (sec Mpc / km)?!! We'll have to use some conversion factors to transform these units into a comprehensible age.

Using the relationships

1 year = 3.15 x 10⁷ sec
1 Mpc = 3.09 x 10¹⁹ km

we find that

                    1   sec Mpc (3.09 x 10¹⁹ km)     (1 year)
 Age of universe = ---- ------- ----------------- ----------------
                    Ho    km         1 Mpc        (3.15 x 10⁷ sec)

Note that the units of sec, Mpc and km all cancel, leaving an age in years.

Calculate the age of the universe, using the numerical value for H_o from Section III. Enter your result in Section IV, Part A of your lab sheet:

    Age of universe = ________________ years.   (Constant expansion.)

Does your answer seem reasonable? How does this compare to the presumed ages of the oldest globular clusters? Write your answer on your lab sheet in the space provided.

It is unlikely the Hubble constant has been constant over the lifetime of the universe. The universe has a lot of mass, and gravity tries to pull all that mass together. Gravity slows the expansion, just as a ball thrown vertically upwards decelerates from the gravitational pull of the earth. The age of the universe computed above corresponds to an "empty universe," one in which the average density of the universe (

) is zero. Should the true age of the universe be larger or smaller than this?

The rate at which the expansion has slowed over time depends on the average density of the universe. The higher the density, the greater the deceleration. If the density is lower than some critical value we call _c, the universe will continue to expand forever. It will get cold and dark. If the density is higher than _c the gravitational attraction will eventually halt the expansion, then begin a contraction toward what has been dubbed the "Big Crunch." If is just equal to _c the expansion will slow to a stop, but only after an infinite amount of time has passed.

Cosmologists often speak in terms of the density parameter _o, defined as

_o =

_c.

Calculating the density of the universe is no simple task. The universe contains perhaps ten times more matter than the luminous matter we can directly detect. We know the "dark matter" is there because we can see its gravitational effects. Most cosmologists believe that baryons ("normal matter") comprise at most 10% of all matter (i.e. _b 0.1). The other 90% is thought to be composed of "exotic" particles we cannot easily detect.

The age of the universe you calculated above corresponds to the case when _o = 0. Suppose the average density of the universe is equal to the critical density, so that _o = 1. In that case, it can be shown that the age of the universe is given by

                        2   
    Age of universe  = ----.     (_o = 1)
                       3 Ho

Multiply your previous result for the age of the universe by 2/3 to calculate the revised age of the universe for the case when

_o = 1. Write your answer on your lab sheet in Part IV, Section B:

    Age of universe = ________________ years.   (_o = 1)

How does this compare to the presumed ages of the oldest globular clusters? What can you infer about the actual value of

_o by comparing the two ages you have calculated? How does your evaluation of

_o compare to what you learned in class? Write your answers on your lab sheet in the spaces provided.

On to A note on scientific error

Back to Hubble's Law and the current Hubble constant

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