4.2. The luminosity distance and gravitational lensing

Before proceeding to discuss possible constraints on from gravitational lensing in this section and high redshift supernovae in the next, let us introduce a quantity which plays a crucial role in these discussions, namely the luminosity distance d_L(z) up to a given redshift z. Consider an object of absolute luminosity located at a coordinate distance r from an observer at r = 0. Light emitted by the object at a time t is received by the observer at t = t₀, t and t₀ being related by the cosmological redshift 1 + z = a(t₀) / a(t). The luminosity flux reaching the observer is

(22)

where d_L is the luminosity distance to the object [142]

(23)

The luminosity distance d_L depends sensitively upon both the spatial curvature and the expansion dynamics of the universe. To demonstrate this we determine d_L using the expression for the coordinate distance r obtained by setting ds² = 0 in (2), resulting in

(24)

which gives

(25)

where = ₀^t cdt / a(t).

Furthermore, since dz / dt = - (1 + z)H(z), we get

(26)

where h(z) = H(z) / H₀ is defined in (18), and, in a universe with several components

(27)

Substituting (27) and (26) in (23) we get the following expression for the luminosity distance in a multicomponent universe with a cosmological term [26]

(28)

where

(29)

and S(x) is defined as follows: S(x) = sin(x) if = 1 (_total > 1), S(x) = sinh(x) if = - 1 (_total < 1), S(x) = x if = 0 (_total = 1).

Figure 4. The luminosity distance dL (in units of H0-1) is shown as a function of cosmological redshift z for flat cosmological models with a cosmological constant m + = 1. Heavier lines correspond to larger values of m. For comparison we also show (dashed line) the angular size in a flat de Sitter universe ( = 1).

Before we turn to applications, let us consider a simple example which provides us with an insight into the role played by the luminosity distance d_L in cosmology. In a spatially flat universe the expression for d_L simplifies considerably, so that we get for the matter dominated model (a t^2/3):

(30)

On the other hand in de Sitter space (a exp(H₀ t))

(31)

Comparing (30) and (31) we find d_L^DS(z) > d_L^MD(z), which means that an object located at a fixed redshift will appear brighter in an Einstein-de Sitter universe than it will in de Sitter space (equivalently in the steady state model). ⁽⁵⁾ This is also true for a two component universe consisting of matter and a cosmological constant as demonstrated in Fig 4. In a spatially flat universe the presence of a -term increases the luminosity distance to a given redshift, leading to interesting astrophysical consequences. Since the physical volume associated with a unit redshift interval increases in models with > 0, the likelihood that light from a quasar will encounter a lensing galaxy is larger in such models. Consequently the probability that a quasar is lensed by intervening galaxies increases appreciably in a dominated universe, and can be used as a test to constrain the value of [72, 71, 192]. Following [73, 26, 37] we give below the probability of a quasar at redshift z_s being lensed relative to the fiducial Einstein-de Sitter model (_m = 1)

(32)

where d (z₁, z₂) is a generalization of the angular distance d_A = d_L(1 + z)^-2 discussed in Section 4.5:

(33)

where

(34)

and S(₁₂) is defined as follows, S(₁₂) = sin(₁₂) if = 1 (_total > 1), S(₁₂) = sinh(₁₂) if = -1 (_total < 1), S(₁₂) = ₁₂ if = 0 (_total = 1). In Fig 5 we show the lensing probability P(lens) for the spatially flat universe _m + = 1. A large increase in the lensing probability over the fiducial _m = 1 value is clearly seen in models with low _m (high ). (For a broader analysis of parameter space see [26].)

Figure 5. The lensing probability P(lens) evaluated relative to the fiducial case m = 1 is shown as a function of for flat cosmological models m + = 1. The source redshift is taken at zs = 1, 2, 3 respectively.

Turning now to the observational situation, at the time of writing the best observational estimates give a 2 upper bound < 0.66 obtained from multiple images of lensed quasars [111, 112, 136]. Since radio sources are not plagued by some of the systematic errors arising in an optical search (notably extinction in the lens galaxy and the quasar discovery process) a search involving radio selected lenses can yield useful complementary information to optical searches [60]. Recent work by Falco et al (1998) gives < 0.73 which is only marginally consistent with optical estimates, a combined analysis of optical and radio data yields a slightly more conservative upper bound < 0.62 at the 2 level (for flat universes) [60]. (Constraints on from both lensing and Type 1a Supernovae are discussed in [201]; also see next section. An interesting new method of constraining from weak lensing in clusters is discussed in [67], also see [12] and section 4.6.) Improved understanding of statistical and systematic uncertainties combined with new surveys and better quality data promise to make gravitational lensing a powerful technique for constraining cosmological parameters and cosmological world models.

⁵ For instance a galaxy at redshift z = 3 will appear 9 times brighter in a flat matter dominated universe than it will in de Sitter space (see Fig 4). Back.