Accelerated cosmic expansion in a scalar-field universe

We consider here a spherically symmetric but inhomogeneous universe filled with a massless scalar field. The model obeys two constraints. The first one is that the gradient of the scalar field is timelike everywhere. The second constraint is that the radial coordinate basis vector is a unit vector field in the comoving coordinate system. We find that the resultant dynamical solutions compose a one-parameter family of self-similar models which is known as the Roberts solution. The solutions are divided into three classes. The first class consists of solutions with only one spacelike singularity in the synchronous-comoving chart. The second class consists of solutions with two singularities which are null and spacelike, respectively. The third class consists of solutions with two spacelike singularities which correspond to the big bang and big crunch, respectively. We see that, in the first case, a comoving volume exponentially expands as in an inflationary period; the fluid elements are accelerated outwards from the symmetry center, even though the strong energy condition is satisfied. This behavior is very different from that observed in the homogeneous and isotropic universe in which the fluid elements would move outwards with deceleration, if the strong energy conditions are satisfied. We are thus able to achieve the accelerated expansion of the universe for the models considered here, without a need to violate the energy conditions. The cosmological features of the models are examined in some detail.

Figure 1

The singularities and Hubble horizon of the analytically extended solution are depicted in $(T,X)$ plane.

Figure 2

The conformal diagram of the analytically extended solution is depicted. This is similar to the upper half of the Minkowski spacetime. Note that the comoving lines, $r=\mathrm{\text{constant}}$, enter into the future null infinity. This means that the elements of stiff fluid are accelerated to the speed of light asymptotically.

Figure 3

The singularities and Hubble horizon for ${\alpha }^2=1/2$ are depicted in the $(T,X)$ plane. There is a central null singularity which is not naked.

Figure 4

The conformal diagram of the analytically extended solution of ${\alpha }^2=1/2$ is depicted. The comoving lines, $r=\mathrm{\text{constant}}$, enter into the timelike infinity.

Figure 5

The singularities and Hubble horizon for $0<{\alpha }^2<1/2$ are depicted in the $(T,X)$ plane.

Figure 6

The conformal diagram of the analytically extended solution of ${\alpha }^2<1/2$ is depicted. There are both the big bang and big crunch singularities, although the topology of the $t=\mathrm{\text{constant}}$ hypersurface is ${\mathbf{R}}^3$.

Figure 7

In the case of the $C^1$-extended solution, the big bang singularity, cosmological horizon, and light ray are depicted in the $(T,X)$ plane.

Figure 8

The conformal diagram of the $C^1$ maximally extended solution is depicted. This solution describes an expanding spherical void in the universe filled with a massless scalar-field.