**2.3. Critical density and the return of the age problem**

One of the things that cosmologists most want to measure accurately is
the total density
of the universe. This is most often expressed in units of the density needed to
make the universe flat, or *k* = 0. Taking the Friedmann equation
for a *k* = 0 universe,

(29) |

we can define a critical density
_{c},

(30) |

which tells us, for a given value of the Hubble constant
*H*_{0}, the energy density of
a Euclidean FRW space. If the energy density is greater than critical,
>
_{c},
the universe is closed and has a positive curvature (*k* = + 1). In
this case, the
universe also has a finite lifetime, eventually collapsing back on
itself in a "big crunch". If
<
_{c},
the universe is open, with negative curvature, and has an
infinite lifetime. This is usually expressed in terms of the density
parameter ,

(31) |

There has long been a debate between theorists and observers as to what the
value of is in the
real universe. Theorists have steadfastly maintained that
the only sensible value for
is unity,
= 1. This prejudice
was further
strengthened by the development of the theory of inflation, which solves
several
cosmological puzzles (see Secs. 4.1 and
4.2) and
in fact *predicts* that
will be
exponentially close to unity. Observers,
however, have made attempts to measure
using a variety of
methods, including measuring galactic rotation curves, the velocities of
galaxies orbiting in clusters,
X-ray measurements of galaxy clusters, the velocities and spatial
distribution of galaxies
on large scales, and gravitational lensing. These measurements have
repeatedly pointed to a value of
inconsistent with a
flat cosmology, with
= 0.2 - 0.3 being
a much better fit, indicating an open, negatively curved universe. Until
a few years ago, theorists have resorted to cheerfully ignoring the
data, taking it almost on faith that
= 0.7 in extra
stuff would turn up sooner or later. The theorists were right: new
observations of the cosmic microwave background definitively favor a
flat universe, = 1.
Unsurprisingly, the observationalists were also right: only about 1/3 of
this density appears to be in the form of ordinary matter.

The first hint that something strange was up with the standard cosmology
came from measurements of the colors of stars in globular
clusters. Globular clusters are small, dense groups of 10^{5} -
10^{6} stars which orbit in the halos of most galaxies and are
among the oldest objects in the universe. Their ages are determined by
observation of stellar populations and models of stellar evolution, and
some globular clusters are known to be at least 12 billion years old
[4],
implying that the universe itself must
be at least 12 billion years old. But consider a flat universe
( = 1) filled
with pressureless matter,
*a*^{-3}
and *p* = 0. It is straightforward to
solve the Friedmann equation (12) with *k* = 0 to show that

(32) |

The Hubble parameter is then given by

(33) |

We therefore have a simple expression for the age of the universe
*t*_{0} in terms of the
measured Hubble constant *H*_{0},

(34) |

The fact that the universe has a finite age introduces the concept of a
*horizon*: this
is just how far out in space we are capable of seeing at any given
time. This distance is
finite because photons have only traveled a finite distance since the
beginning of the
universe. Just as in special relativity, photons travel on paths with
proper length *ds*^{2} = *dt*^{2} -
*a*^{2}*d***x**^{2} = 0, so that we can
write the physical distance a photon has
traveled since the Big Bang, or the *horizon size*, as

(35) |

(This is in units where the speed of light is set to *c* = 1.) For
example, in a flat, matter-dominated universe,
*a*(*t*)
*t*^{2/3}, so that the horizon size is

(36) |

This form for the horizon distance is true in general: the distance a
photon travels in time *t* is always about *d* ~ *t*:
effects from expansion simply add a numerical factor
out front. We will frequently ignore this, and approximate

(37) |

Measured values of *H*_{0} are quoted in a very strange
unit of time, a km/s/Mpc, but it is
a simple matter to calculate the dimensionless factor using
1 Mpc 3 ×
10^{19} km,
so that the age of a flat, matter-dominated universe with
*H*_{0} = 71 ± 6 km/s/Mpc is

(38) |

A flat, matter-dominated universe would be younger than the things in it! Something is evidently wrong - either the estimates of globular cluster ages are too big, the measurement of the Hubble constant from from the HST is incorrect, the universe is not flat, or the universe is not matter dominated.

We will take for granted that the measurement of the Hubble constant is
correct, and that
the models of stellar structure are good enough to produce a reliable
estimate of globular
cluster ages (as they appear to be), and focus on the last two
possibilities. An open universe,
_{0} < 1,
might solve the age problem. Figure 1 shows the
age of the universe consistent with the HST Key Project value for
*H*_{0} as a function of the density parameter
_{0}.

We see that the age determined from *H*_{0} is consistent
with globular clusters as old as 12 billion years only for values of
_{0} less
than 0.3 or so. However, as we will see in
Sec. 3, recent measurements of the cosmic
microwave background strongly indicate that we indeed live in a flat
( = 1) universe. So
while a low-density universe might
provide a marginal solution to the age problem, it would conflict with
the CMB. We therefore, perhaps reluctantly, are forced to consider that
the universe might not be matter dominated. In the next section we will
take a detour into quantum field theory seemingly unrelated to these
cosmological issues. By the time we are finished, however, we will have
in hand a different, and provocative, solution to the age problem
consistent with a flat universe.