2.4. Geometrical features of a universe with a cosmological constant
The evolution of the universe has different characteristic features if there exists sources in the universe for which (1 + 3w) < 0. This is obvious from equation (8) which shows that if ( + 3P) = (1 + 3w) becomes negative, then the gravitational force of such a source (with > 0) will be repulsive. The simplest example of this kind of a source is the cosmological constant with w_{} = - 1.
To see the effect of a cosmological constant let us consider a universe with matter, radiation and a cosmological constant. Introducing a dimensionless time coordinate = H_{0} tand writing a = a_{0} q() equation (20) can be cast in a more suggestive form describing the one dimensional motion of a particle in a potential
(24) |
where
(25) |
This equation has the structure of the first integral for motion of a particle with energy E in a potential V(q). For models with = _{R} + _{NR} + _{} = 1, we can take E = 0 so that (dq / d) = [V(q)]^{1/2}. Figure 2 is the phase portrait of the universe showing the velocity (dq / d) as a function of the position q = (1 + z)^{-1} for such models. At high redshift (small q) the universe is radiation dominated and is independent of the other cosmological parameters; hence all the curves asymptotically approach each other at the left end of the figure. At low redshifts, the presence of cosmological constant makes a difference and - in fact - the velocity changes from being a decreasing function to an increasing function. In other words, the presence of a cosmological constant leads to an accelerating universe at low redshifts.
Figure 2. The phase portrait of the universe, with the "velocity" of the universe (dq / d) plotted against the "position" q = (1 + z)^{-1} for different models with _{R} = 2.56 × 10^{-5}h^{-2}, h = 0.5, _{NR} + _{} + _{R} = 1. Curves are parameterized by the value of _{NR} = 0.1, 0.2, 0.3, 0.5, 0.8, 1.0 going from bottom to top as indicated. (Figure adapted from [46].) |
For a universe with non relativistic matter and cosmological constant, the potential in (25) has a simple form, varying as (- a^{-1}) for small a and (- a^{2}) for large a with a maximum in between at q = q_{max} = (_{NR} / 2_{})^{1/3}. This system has been analyzed in detail in literature for both constant cosmological constant [67] and for a time dependent cosmological constant [68]. A wide variety of explicit solutions for a(t) can be provided for these equations. We briefly summarize a few of them.
(26) |
was the one which originally prompted Einstein to introduce the cosmological constant (see section 1.2).
(27) |
This solution smoothly interpolates between a matter dominated universe a(t) t^{2/3} at early stages and a cosmological constant dominated phase a(t) exp(Ht) at late stages. The transition from deceleration to acceleration occurs at z_{acc} = (2_{} / _{NR})^{1/3} - 1, while the energy densities of the cosmological constant and the matter are equal at z_{m} = (_{} / _{NR})^{1/3} - 1.
The presence of a cosmological constant also affects other geometrical parameters in the universe. Figure 3 gives the plot of d_{A}(z) and d_{L}(z); (note that angular diameter distance is not a monotonic function of z). Asymptotically, for large z, these have the limiting forms,
(28) |
The geometry of the spacetime also determines the proper volume of the universe between the redshifts z and z + dz which subtends a solid angle d in the sky. If the number density of sources of a particular kind (say, galaxies, quasars, ...) is given by n(z), then the number count of sources per unit solid angle per redshift interval should vary as
(29) |
Figure 4 shows (dN / ddz); it is assumed that n(z) = n_{0}(1 + z)^{3}. The y-axis is in units of n_{0} H_{0}^{-3}.
Figure 4. The figure shows (dN / ddz): it is assumed that n(z) = n_{0}(1 + z)^{3}. The y-axis is in units of n_{0}H_{0}^{-3}. Each curve is labelled by (_{NR}, _{}). |