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.  In co-moving, spherical coordinates, this metric tensor leads to the well known line element
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 t0, is a three-dimensional hypersphere of constant scalar curvature K(t0) = k / a2(t0). 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
where Rµ is the Ricci tensor, R is the scalar curvature, and Tµ 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)
where uµis the four-velocity of the fluid, is its energy density, and p is the fluid pressure. In Eqs. (2) and (3), gµ is relative to the Cartesian coordinates (x0 = ct, x1 = x, x2 = y, x3 = z). Under the restrictions imposed by the cosmological principle, the field equations (2) reduce to the Friedmann equations for the cosmic scale factor
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
where vr 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 H0, called the Hubble constant. 
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 H2, we can rewrite this expression as
and make the following definition:
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
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 + 3p > 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
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
Collectively, the Hubble constant H0 and the present values of the density parameter 0 and the deceleration parameter q0 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.
|50 < H0 < 100 km . s-1 . Mpc-1|
|Open||< 1||m < c||q0 < 1/2|
|Closed||> 1||m > c||q0 > 1/2|
|Flat||= 1||m = c||q0 = 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.  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 , 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.