Topological implications of inhomogeneity

The approximate homogeneity of spatial sections of the Universe is well supported observationally, but the inhomogeneity of the spatial sections is even better supported. Here, we consider the implications of inhomogeneity in dust models for the connectedness of spatial sections at early times. We consider a nonglobal Lemaître-Tolman-Bondi (LTB) model designed to match observations, a more general, heuristic model motivated by the former, and two specific, global LTB models. We propose that the generic class of solutions of the Einstein equations projected back in time from the spatial section at the present epoch includes subclasses in which the spatial section evolves (with increasing time) smoothly (i) from being disconnected to being connected, or (ii) from being simply connected to being multiply connected, where the coordinate system is comoving and synchronous. We show that (i) and (ii) each contain at least one exact solution. These subclasses exist because the Einstein equations allow nonsimultaneous big bang times. The two types of topology evolution occur over time slices that include significantly postquantum epochs if the bang time varies by much more than a Planck time. In this sense, it is possible for cosmic topology evolution to be “mostly” classical.

Figure 1

Coordinate-space illustration of the spatial, comoving section $H^3\setminus \mathcal{V}$ in the empirical solution in Ref. 12 at $t=-1\mathrm{Gyr}$, discussed in Sec. 2. The closed three-dimensional ball $\mathcal{V}$ (8) consists of coordinate space that is not part of the physically defined, spatial 3-manifold. The boundary $\partial \mathcal{V}$ has zero metrical area $4\pi R^2$ and corresponds to a spatial section through the initial singularity, i.e., $\partial \mathcal{V}\subset \mathcal{V}\Rightarrow \partial \mathcal{V}\subset H^3\setminus \mathcal{V}$. The $r=3.7\mathrm{Gpc}$ 2-sphere shows the limit of the authors’ fit to observations. We extrapolate this to arbitrarily large $r$. The physically defined spatial section $H^3\setminus \mathcal{V}$ is shaded in grey up to an arbitrary cutoff radius.

Figure 2

Coordinate-space illustration of part of the spatial, comoving section at $t=-3\sigma$ of the spacetime solution suggested in Sec. 3 to be more general than the empirical solution in Ref. 12. The physical (metrical) spatial section at coordinate time $t=-3\sigma$ is the spatially disconnected space $\cup \{{\mathcal{W}}_i\}$, shown here for $1\le i\le 3$. The boundary $\cup \{\partial {\mathcal{W}}_i\}$ has zero metrical area and corresponds to a spatial section through the initial singularity.

Figure 3

Example of $t_{\mathrm{B}}$-inhomogeneous positively curved LTB solution (15, 16) that is born as disconnected spatial sections that smoothly merge together via the initial singularity (cf. Fig. 7(a) of Ref. 22), showing a finite part of the comoving spatial section, which is of infinite length in the $r$ direction. The Universe exists (has emerged from the big bang singularity) in the shaded region (excluding the singularity itself, appearing as a sinusoid here). The thin horizontal line shows a spatial section of the Universe at $t=-0.42h^{-1}\mathrm{Gyr}$, during which some parts of the Universe exist, and others do not yet exist.

Figure 4

Density $\rho$ of the solution shown in Fig. 3. Values are plotted for the ranges of $r$ where the Universe exists, i.e., $t>t_{\mathrm{B}}(r)$. Values increasing arbitrarily are shown to limits that depend on numerical implementation details and avoid obscuring the legends. A late epoch, near recollapse, is shown by a thinner curve in this figure and those following.

Figure 5

As for Fig. 4, a Hubble-like parameter $|H|$ (18). The $H$ parameter of the late epoch (near recollapse, thinner curve) has negative values of $H$.

Figure 6

As for Fig. 4, the Ricci scalar ${}^{3}R/6$ (17).

Figure 7

As for Fig. 4, the areal radius $R(t,r)$.

Figure 8

As for Fig. 4, the radial metric component $\sqrt{g_{rr}(t,r)}$.

Figure 9

As for Fig. 4, the radial proper length $d(t,r)$ (19) at several preconnection and postconnection epochs $t$. The two thick curves match epochs shown in the previous figures.

Figure 10

As for Fig. 3, example of inhomogeneous positively curved LTB solution with nonsimultaneous (i.e., nonconstant) $t_{\mathrm{B}}(r)$ that is born with a simply connected spatial section that smoothly connects to itself via the initial singularity, becoming multiply connected ($S^2\times S^1$), shown in the universal covering space. One copy of the fundamental domain (FD) lies between the two vertical lines.