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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 (rho + 3P) = (1 + 3w)rho becomes negative, then the gravitational force of such a source (with rho > 0) will be repulsive. The simplest example of this kind of a source is the cosmological constant with wLambda = - 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 tau = H0 tand writing a = a0 q(tau) equation (20) can be cast in a more suggestive form describing the one dimensional motion of a particle in a potential

Equation 24 (24)


Equation 25 (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 Omega = OmegaR + OmegaNR + OmegaLambda = 1, we can take E = 0 so that (dq / dtau) = [V(q)]1/2. Figure 2 is the phase portrait of the universe showing the velocity (dq / dtau) as a function of the position q = (1 + z)-1 for such models. At high redshift (small q) the universe is radiation dominated and qdot 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 qdot 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

Figure 2. The phase portrait of the universe, with the "velocity" of the universe (dq / dtau) plotted against the "position" q = (1 + z)-1 for different models with OmegaR = 2.56 × 10-5h-2, h = 0.5, OmegaNR + OmegaLambda + OmegaR = 1. Curves are parameterized by the value of OmegaNR = 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 (- a2) for large a with a maximum in between at q = qmax = (OmegaNR / 2OmegaLambda)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.

The presence of a cosmological constant also affects other geometrical parameters in the universe. Figure 3 gives the plot of dA(z) and dL(z); (note that angular diameter distance is not a monotonic function of z). Asymptotically, for large z, these have the limiting forms,

Equation 28 (28)

Figure 3a
Figure 3b

Figure 3. The left panel gives the angular diameter distance in units of cH0-1 as a function of redshift. The right panel gives the luminosity distance in units of cH0-1 as a function of redshift. Each curve is labelled by (OmegaNR, OmegaLambda).

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 dOmega 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

Equation 29 (29)

Figure 4 shows (dN / dOmegadz); it is assumed that n(z) = n0(1 + z)3. The y-axis is in units of n0 H0-3.

Figure 4

Figure 4. The figure shows (dN / dOmegadz): it is assumed that n(z) = n0(1 + z)3. The y-axis is in units of n0H0-3. Each curve is labelled by (OmegaNR, OmegaLambda).

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