1. The XCDM parametrization

In the XCDM parametrization the dark energy interacts only with itself and gravity, the dark energy density _X(t) > 0 is approximated as a function of world time alone, and the pressure is written as

(43)

an expression that has come to be known as the cosmic equation of state. ⁽⁴⁴⁾ Then the local energy conservation equation (9) is

(44)

If w_X is constant the dark energy density scales with the expansion factor as

(45)

If w_X < - 1/3 the dark energy makes a positive contribution to /a (Eq. [8]). If w_X = - 1/3 the dark energy has no effect on , and the energy density varies as _X 1/a², the same as the space curvature term in ² / a² (Eq. [11]). That is, the expansion time histories are the same in an open model with no dark energy and in a spatially-flat model with w_X = - 1/3, although the spacetime geometries differ. ⁽⁴⁵⁾ If w_X < - 1 the dark energy density is increasing. This is quite a step from the thought that the dark energy density is small because it has been rolling to zero for a long time, but the case has found a context (Maor et al., 2002).

Equation (45) with constant w_X has the great advantage of simplicity. An appropriate generalization for the more precise measurements to come might be guided by the idea that the dark energy density is close to homogeneous, spatial variations rearranging themselves at close to the speed of light, as in the scalar field models discussed below. Then for most of the cosmological tests we have an adequate general description of the dark energy if we let w_X be a free function of time. ⁽⁴⁶⁾ In scalar field pictures w_X is derived from the field model; it can be a complicated function of time even when the potential is a simple function of the scalar field.

The analysis of the large-scale anisotropy of the 3 K cosmic microwave background radiation requires a prescription for how the spatial distribution of the dark energy is gravitationally related to the inhomogeneous distribution of other matter and radiation (Caldwell et al., 1998). In XCDM this requires at least one more parameter, an effective speed of sound, with c²_sX > 0 (for stability, as discussed in Sec. II.B), in addition to w_X.

⁴⁴ Other parametrizations of dark energy are discussed by Hu (1998) and Bucher and Spergel (1999). The name, XCDM, for the case w_X < 0 in Eq. (43), was introduced by Turner and White (1997). There is a long history in cosmology of applications of such an equation of state, and the related evolution of ; examples are Canuto et al. (1977), Lau (1985), Huang (1985), Fry (1985), Hiscock (1986), Özer and Taha (1986), and Olson and Jordan (1987). See Ratra and Peebles (1988) for references to other early work on a time-variable and Overduin and Cooperstock (1998) and Sahni and Starobinsky (2000) for reviews. More recent discussions of this and related models may be found in John and Joseph (2000), Zimdahl et al. (2001), Dalal et al. (2001), Gudmundsson and Björnsson (2002), Bean and Melchiorri (2002), Mak, Belinchón, and Harko (2001), and Kujat et al. (2002), through which other recent work may be traced. Back.

⁴⁵ As discussed in Sec. IV, it appears difficult to reconcile the case w_X = -1/3 with the Type Ia supernova apparent magnitude data (Garnavich et al., 1998; Perlmutter et al., 1999a). Back.

⁴⁶ The availability of a free function greatly complicates the search for tests as opposed to curve fitting! This is clearly illustrated by Maor et al. (2002). For more examples see Perlmutter, Turner, and White (1999b) and Efstathiou (1999). Back.