The basic tenet that governs cosmology is known as *the cosmological
principle*. This principle states that, on large scales, the present
universe is homogeneous and isotropic. Homogeneity means that the
properties of the universe are the same everywhere in the universe;
and isotropy means that from every point, the properties of the
universe are the same in every direction. It can be shown that the
cosmological principle alone requires that the metric tensor of the
universe must take the form of the Robertson-Walker metric.
[1] In
co-moving, spherical coordinates, this metric tensor leads to the
well known line element

(1) |

where the dimensionless function *a*(*t*) is called the cosmic
scale factor. This line element describes an expanding (or contracting)
universe that, at the present (or any) instant in time
*t*_{0}, is a
three-dimensional hypersphere of constant scalar curvature
*K*(*t*_{0}) = *k* /
*a*^{2}(*t*_{0}). The parameter *k*
represents the sign of this constant which
can either be positive (*k* = + 1 m^{-2}), negative
(*k* = - 1 m^{-2}), or zero (*k* = 0).

The dynamics of the universe is governed by the Einstein field equations

(2) |

where
*R*_{µ} is
the Ricci tensor, *R* is the scalar curvature, and
*T*_{µ} is
the stress-energy tensor. On large scales, the stress-energy
tensor of the universe is taken to be that of a perfect fluid (since
homogeneity and isotropy imply that there is no bulk energy transport)

(3) |

where *u*_{µ}is the four-velocity of the fluid,
is its energy
density, and *p* is the fluid pressure. In Eqs. (2) and (3),
*g*_{µ} is
relative to the Cartesian coordinates (*x*^{0} = *ct*,
*x*^{1} = *x*,
*x*^{2} = *y*, *x*^{3} = *z*).
Under the restrictions imposed by the cosmological principle, the field
equations (2) reduce to the Friedmann equations for the cosmic scale
factor

(4) |

(5) |

where the dot notation represents a derivative with respect to time, as is customary. Observationally, it is known that the universe is expanding. The expansion of the universe follows the Hubble law

(6) |

where *v*_{r} is the speed of recession between two points,
*d* is the proper distance between these points, and
*H*(
/*a*) is
called the *Hubble parameter*. One of the key features of the Hubble
law is that, at any given instant, the speed of recession is directly
proportional to the distance. Therefore, by analyzing the Doppler shift
in the light from a distant source we can infer its distance provided
that we know the present value of the Hubble parameter
*H*_{0}, called the *Hubble constant*.
[2]

One of the principal questions that the field of cosmology hopes to
answer concerns the ultimate fate of the universe. Will the universe
expand forever, or will the expansion halt and be followed by a
contraction? This question of the long-term fate of the expansion is
closely connected to the sign of *k* in the Friedmann equation
(5). General relativity teaches us that the curvature of spacetime is
determined by the density of matter and energy. Therefore, the two terms
on the right-hand-side of Eq. (5) are not independent. The value of the
energy density will determine the curvature of spacetime and,
consequently, the ultimate fate of the expansion. Recognizing that the
left-hand-side of Eq. (5) is *H*^{2}, we can rewrite this
expression as

(7) |

and make the following definition:

(8) |

called the density parameter.

Since the sum of the density and curvature terms in Eq. (7) equals
unity, the case for which
< 1 corresponds
to a negative curvature term requiring *k* = - 1
m^{-2}. The solution for negative
curvature is such that the universe expands forever with excess velocity
_{t=} > 0. This latter
case is referred to as an *open*
universe. The case for which corresponds to a positive curvature term
requiring *k* = + 1 m^{-2}. The solution for positive
curvature (a *closed* universe) is such that the expansion
eventually halts and becomes a universal contraction leading to what is
known as the *big crunch*. Finally, the case for which
= 1 corresponds to zero
curvature (a *flat* universe) requiring *k* = 0. The solution
for a flat universe is the critical case that lies on the boundary
between an
open and closed universe. In this case, the universe expands forever,
but the rate of expansion approaches zero asymptotically,
_{t=} = 0. The value of the
energy density for which
= 1 is called the
*critical density* _{c} , given by

(9) |

The density parameter, then, is the ratio of the energy density of the
universe to the critical density
=
/
_{c}.

There is one final parameter that is used to characterize the universal
expansion. Notice that Eq. (4) expresses the basic result that in a
matter-dominated universe (in which
+ 3*p*
> 0) the expansion should be decelerating,
< 0, as a result of
the collective
gravitational attraction of the matter and energy in the universe. This
behavior is characterized by the *deceleration parameter*

(10) |

The matter in the present universe is very sparse, so that it is
effectively noninteracting (i.e., dust). Therefore, it is generally
assumed that the fluid pressure, *p*, is negligible compared to the
energy density. Under these conditions, Eq. (4) shows that the
deceleration parameter has a straightforward relationship to the density
parameter

(11) |

Collectively, the Hubble constant *H*_{0} and the present
values of the density parameter
_{0} and the
deceleration parameter *q*_{0} are
known as the *cosmological parameters*. These parameters are chosen,
in part, because they are potentially measurable. The range of values
that correspond to the different fates of the universe, within this
traditional framework, are summarized in Table 1.

Model | Parameters | ||

50 < H_{0} < 100 km
^{.} s^{-1 . } Mpc^{-1} |
|||

Open | < 1 | _{m} <
_{c} |
q_{0} < 1/2 |

Closed | > 1 | _{m} >
_{c} |
q_{0} > 1/2 |

Flat | = 1 | _{m} =
_{c} |
q_{0} = 1/2 |

The understanding of cosmology, as outlined above, left several
questions unanswered, including the fate of the universal expansion. It
was entirely possible to formulate reasonable arguments for whether or
not the universe is open, closed, or flat that covered all three
possibilities. The observational data has always suggested that the
density of visible matter is insufficient to close the universe and
researchers choosing to side with the data could easily take the
position that the universe is open. However, it has been known for
several decades that a substantial amount of the matter in the universe,
perhaps even most of it, is not visible. The existence of large amounts
of *dark matter* can be inferred from its gravitational effects both on
and within galaxies.
[3]
Therefore, the prospects of dark matter
(and neutrino mass) rendered any conclusion based solely on the amount of
visible matter premature. Einstein's view was that the universe is
closed, apparently for reasons having to do with Mach's principle
[4],
and many researchers preferred this view as well for
reasons that were sometimes more philosophical than scientific. Then,
there were also hints that the universe may be flat; consequentially,
many researchers believed that this was most likely true.

Belief that the universe is flat was partly justified by what is known
in cosmology as the *flatness problem* having to do with the
apparent need of the universe to have been exceedingly close to the
critical density shortly after the big bang. The fate of the universe
and the flatness problem are just two of several puzzles that emerge
from this standard model of the universe. Another important puzzle has
to do with the existence, or nonexistence, of the *cosmological
constant*,
. It turns out that
plays a very
important role in our story, and its story must be told before we can
explain how cosmologists have pinned down some of the basic properties
of the universe.