Quantum cosmology of a conformal multiverse

This paper studies the cosmology of a homogeneous and isotropic spacetime endorsed with a conformally coupled massless scalar field. We find six different solutions of the Friedmann equation that represent six different types of universes, and all of them are periodically distributed along the complex time axis. From a classical point of view, they are then isolated, separated by Euclidean regions that represent quantum mechanical barriers. Quantum mechanically, however, there is a nonzero probability for the state of the universes to tunnel out through a Euclidean instanton and suffer a sudden transition to another state of the spacetime. We compute the probability of transition for this and other nonlocal processes like the creation of universes in entangled pairs and, generally speaking, in multipartite entangled states. We obtain the quantum state of a single universe within the formalism of the Wheeler-DeWitt equation and give the semiclassical state of the universes that describes the quantum mechanics of a scalar field propagating in a de Sitter background spacetime. We show that the superposition principle of the quantum mechanics of matter fields alone is an emergent feature of the semiclassical description of the universe that is not valid, for instance, in the spacetime foam. We use the third quantization formalism to describe the creation of an entangled pair of universes with opposite signs of the momentum conjugated to the scale factor. Each universe of the entangled pair represents an expanding spacetime in terms of the Wentzel-Kramers-Brillouin (WKB) time experienced by internal observers in their particle physics experiments. We compute the effective value of the Friedmann equation of the background spacetime of the two entangled universes, and thus, the effect that the entanglement would have in their expansion rates. We analyze as well the effects of the interuniversal entanglement in the properties of the scalar fields that propagate in each spacetime of the entangled pair. We find that the largest modes of the scalar field are unaware of the entanglement between the universes, but the effects can be significant for the lowest modes, allowing us to compute, in principle, detailed observational imprints of the multiverse in the properties of a single universe like ours.

Figure 1

Lorentzian and Euclidean regions for the potential $V(a)=a^2-H^2{a}^4$ in (20), with $C=2ϵ$. For the value, $4H^{2}C>1$, there is no forbidden region. For the value, $4H^{2}C<1$, there are two classically allowed regions separated by a quantum barrier.

Figure 2

Solution (33) of the Friedmann equation for $A_0^2=\pm 1$ and, $4H^{2}C>1$.

Figure 3

Solution (34) of the Friedmann equation for $A_0^2=\pm 1$ and, $4H^{2}C<1$.

Figure 4

Solution (35) of the Friedmann equation for $A_0^2=\pm 1$ and, $4H^{2}C=1$.

Figure 5

A cyclic universe can be connected with a de Sitter like universe by means of the Euclidean instanton (48).

Figure 6

The creation of a pair of cyclic universes from a double instanton.

Figure 7

The creation of a large parent universe from a baby universe.

Figure 8

A double Euclidean instanton can be formed by matching two single Euclidean instantons.

Figure 9

The creation of a pair of entangled universes from nothing, i.e., from a double Euclidean instanton.

Figure 10

The creation of a multipartite entangled state of $N$ universes from a $N$-Euclidean instanton (see Fig. 11).

Figure 11

A $N$-Euclidean instanton.

Figure 12

A multiple Euclidean instanton can also be formed by gluing instantons of different types.

Figure 13

Energy ($ϵ$), work ($w$), and heat ($q$) densities of entanglement, Eq. (159).

Figure 14

Energy density of entanglement [see Eqs. (156)–(159)] for different values of the spatial mode of the scalar field.

Figure 15

Spectrum of quantum fluctuations for the vacuum state, ${\delta }_v$ given by (164) with, $L_{\mathrm{ph}}=\frac{a}{n_{\mathrm{ph}}}$, and for the thermal state (144) derived from the entanglement between the two parent spacetimes where the scalar fields propagate.