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The most symmetric vacuum solution to Einstein's equation, of course, is the flat spacetime. If we now add the cosmological constant as the only source of curvature in Einstein's equation, the resulting spacetime is also highly symmetric and has an interesting geometrical structure. In the case of a positive cosmological constant, this is the de Sitter manifold and in the case of negative cosmological constant, it is known as anti-de Sitter manifold. We shall now discuss some features of the former, corresponding to the positive cosmological constant.

To understand the geometrical structure of the de Sitter spacetime, let us begin by noting that a spacetime with the source Tba = rhoLambda deltaab must have 3-dimensional section which are homogeneous and isotropic. This will lead us to the Einstein's equations for a FRW universe with cosmological constant as source

Equation 130 (130)

This equation can be solved with any of the following three forms of (k, a(t)) pair. The first pair is the spatially flat universe with (k = 0, a = eH t). The second corresponds to spatially open universe with (k = - 1, a = H-1sinh H t) and the third will be (k = + 1, a = H-1cosh H t). Of these, the last pair gives a coordinate system which covers the full de Sitter manifold. In fact, this is the metric on a 4-dimensional hyperboloid, embedded in a 5 dimensional Minkowski space with the metric

Equation 131 (131)

The equation of the hyperboloid in 5-D space is

Equation 132 (132)

We can introduce a parametric representation of the hyperbola with the four variables (tau, chi, theta, phi) where

Equation 133 (133)

This set, of course, satisfies (132). Using (131), we can compute the metric induced on the hyperboloid which - when expressed in terms of the four coordinates (tau, chi, theta, phi) - is given by

Equation 134 (134)

This is precisely the de Sitter manifold with closed spatial sections.

All the three forms of FRW universes with k = 0, ± 1 arise by taking different cuts in this 4-dimensional hyperboloid embedded in the 5-dimensional spacetime. Since two of these dimensions (corresponding to the polar angles theta and phi) merely go for a ride, it is more convenient (for visualization) to work with a 3-dimensional spacetime having the metric

Equation 135 (135)

instead of the 5-dimensional metric (131). Every point in this 3-dimensional space corresponds to a 2-sphere whose coordinates theta and phi are suppressed for simplicity. The (1 + 1) de Sitter spacetime is the 2-dimensional hyperboloid [instead of the four dimensional hyperboloid of (132)] with the equation

Equation 136 (136)

embedded in the 3-dimensional space with metric (135). The three different coordinate systems which are natural on this hyperboloid are the following:

All these metrics have an apparent time dependence. But, in the absence of any source other than cosmological constant, there is no preferred notion of time and the spacetime manifold cannot have any intrinsic time dependence. This is indeed true, in spite of the expansion factor a(t) ostensibly depending on time. The translation along the time direction merely slides the point on the surface of the hyperboloid. [This is obvious in the coordinates (k = 0, a propto eH t) in which the time translation t rightarrow t + epsilon merely rescales the coordinates by (exp H epsilon).]

The time independence of the metric can be made explicit in another set of coordinates called `static coordinates'. To motivate these coordinates, let us note that a spacetime with only cosmological constant as the source is certainly static and possesses spherical symmetry. Hence we can also express the metric in the form

Equation 140 (140)

where nu and lambda are functions of r. The Einstein's equations for this metric has the solution enu = e-lambda = (1 - H2 r2) leading to

Equation 141 (141)

This form of the metric makes the static nature apparent. This metric also describes a hyperboloid embedded in a higher dimensional flat space. For example, in the (1 + 1) case (with theta, phi suppressed) this metric can be obtained by the following parameterization of the hyperboloid in equation (136):

Equation 142 (142)

The key feature of the manifold, revealed by equation (141) is the existence of a horizon at r = H-1. It also shows that t is a time-like coordinate only in the region r < H-1.

The structure of the metric is very similar to the Schwarzschild metric:

Equation 143 (143)

Both the metrics (143) and (141) are spherically symmetric with g00 = - (1/g11). Just as the Schwarzschild metric has a horizon at r = 2M (indicated by g00 rightarrow 0, g11 rightarrow infty), the de Sitter metric also has a horizon at r = H-1. From the slope of the light cones (dt / dr) = ± (1 - H2 r2)-1 [corresponding to ds = 0 = dtheta = dphi in (142)] it is clear that signals sent from the region r < H-1 cannot go beyond the surface r = H-1.

This feature, of course, is independent of the coordinate system used. To see how the horizon in de Sitter universe arises in the FRW coordinates, let us recall the equation governing the propagation of light signals between the events (t1, r1) and (t, r):

Equation 144 (144)

Consider a photon emitted by an observer at the origin at the present epoch (r1 = 0, t1 = t0). The maximum coordinate distance xH reached by this photon as t rightarrow infty is determined by the equation

Equation 145 (145)

If the integral on the right hand side diverges as t rightarrow infty, then, in the same limit, xH rightarrow infty and an observer can send signals to any event provided (s)he waits for a sufficiently long time. But if the integral on the right hand side converges to a finite value as t rightarrow infty, then there is a finite horizon radius beyond which the observer's signals will not reach even if (s)he waits for infinite time. In the de Sitter universe with k = 0 and a(t) = eH t, xH = H-1 e-H t0; the corresponding maximum proper distance up to which the signals can reach is rH = a(t0) xH = H-1. Thus we get the same result in any other coordinate system.

Since the result depends essentially on the behaviour of a(t) as t rightarrow infty, it will persist even in the case of a universe containing both non relativistic matter and cosmological constant. For example, in our universe, we can ask what is the highest redshift source from which we can ever receive a light signal, if the signal was sent today. To compute this explicitly, consider a model with OmegaNR + OmegaLambda = 1. Let us assume that light from an event at (rH, zH) reaches r = 0 at z = 0 giving

Equation 146 (146)

If we take rH to be the size of the horizon, then it also follows that the light emitted today from this event will just reach us at t = infty. This gives

Equation 147 (147)

Equating the two expressions, we get an implicit expression for zH. If OmegaNR = 0.3, the limiting redshift is quite small: zH approx 1.8. This implies that sources with z > zH can never be influenced by light signals from us in a model with cosmological constant [286, 287].

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