#### 2.1. Quantum Multiverse

Let us consider a homogeneous and isotropic spacetime with metric element given by

where

$a\left(t\right)$ is the scale factor and

$d{\mathsf{\Omega}}_{3}^{2}$ is the metric element on the three sphere of unit radius. Let us also consider a scalar field minimally coupled to the spacetime. The Hamiltonian constraint is then given by

where [

7],

${\sigma}^{2}=\frac{3\pi {M}_{P}^{2}}{2}$ and

${H}^{2}\left(\phi \right)=\frac{8\pi}{3{M}_{P}^{2}}V\left(\phi \right)$. Following the canonical procedure of quantization the momenta are then promoted to operators and the Hamiltonian Constraint (

2) transforms into the Wheeler-DeWitt equation, which can be written as [

7]

where

$\varphi \equiv \varphi (a,\phi )$, is the wave function of the universe [

8], the dot means derivative with respect to the scale factor and the prime denotes the derivative with respect to the scalar field. The frequency

$\omega (a,\phi )$ contains the potential terms of the Hamiltonian Constraint (

2). It is given by

The Wheeler-DeWitt Equation (

3) has been written in a way that enhances the formal analogy with the wave equation of a scalar field. The scalar field to be quantized is now the wave function of the universe that propagates in the minisuperspace spanned by the variables,

${q}^{N}=\{a,\phi \}$, with metric element (of the minisuperspace) given by

The third quantization formalism [

4,

5] consist of considering and extending this formal analogy and quantize the wave function of the universe in a similar way as it is done in a quantum field theory. In particular, we can start by considering an action functional from which the wave Equation (

3) can be obtained. It is given by

1
with

Let us notice that in the metric element (

5) as well as in the action (

6) the scale factor formally plays the role of the time like variable. Then, the momentum conjugated to the wave function of the universe,

$\varphi $, is given by

and the Hamiltonian density is then

which essentially is the Hamiltonian of a harmonic oscillator with

time dependent mass,

$M\left(a\right)=a$, and frequency,

$\omega (a,\phi )$, given by (

4).

#### 2.2. Interacting Scheme

We can now pose an scheme of interaction between universes in a parallel way to how is done in quantum mechanics, by considering a total Hamiltonian given by [

9]

where

${\mathcal{H}}_{n}^{\left(0\right)}$ is given by (

9) for the universe

n, and it corresponds to the Hamiltonian of a non-interacting universe. The interaction is described then by the Hamiltonian of interaction,

${\mathcal{H}}_{n}^{I}$. For this let us consider the following quadratic Hamiltonian

with the boundary condition,

${\varphi}_{N+1}\equiv {\varphi}_{1}$. As it is well known in quantum mechanics, we can consider the following Fourier transformation

in terms of which the normal modes the Hamiltonian (

10) turns out to represent

N non-interacting new universes, i.e.,

where

with a new effective value of the frequency given by

with,

${\tilde{H}}_{k}^{2}=\frac{8\pi}{3{M}_{P}^{2}}{\tilde{V}}_{k}(a,\phi )$, and

The final result of the interaction is then an effective modification of the potential of the scalar field. However, it is worth noticing that the classical field equations are not modified because (here

$\dot{\phi}\equiv \frac{d\phi}{dt}$)

The extra term in the potential (

16) entails a shift of the ground state that classically has no influence in the field equations. However, it determines the structure of the quantum vacuum states as well as the global structure of the whole spacetime.

#### 2.3. Modified Properties

Let us now consider several examples where the influence of the interaction among universes may be important. Let us first notice that the interaction among universes is not expected to have a significant influence in a large parent universe like ours. However, it may have a strong effect in small baby universes and thus, in the very early stage of the evolution of the universes. These effects may then propagate along the subsequent evolution of the universe and reach us as small corrections to the expected values of the non-interacting models of the universe (i.e., single universe models).

For concreteness, let us focus on the quartic potential studied in Ref. [

9]. In that case, the interaction among universes induce a landscape structure of different false vacua and one or two true vacua (see

Figure 1). Moreover, new processes of vacuum decay can now be posed including simple processes of vacuum decay as well as double decays that would lead to the formation of entangled pairs of spacetime bubbles. The decaying rate per unit volume between two consecutive levels of the potential is given by

with [

9]

where

m is the mass of the scalar field,

${\lambda}_{\phi}$, is the coupling of the quartic term in the potential, and

$\lambda =\lambda \left(a\right)$ is the coupling function that determines the interaction between the universes (see Equation (

11)). Equation (

19) imposes a restriction on the values of the coupling function

$\lambda $. Let us notice that in order for the vacuum decay to be suppressed for large parent universes like ours, then,

$\frac{{\lambda}^{2}}{{a}^{4}}\to 0$, in the limit of large values of the scale factor. Even though, there are interesting cases fulfilling this condition with possibly observable imprints in the properties of the CMB of a universe like ours.

For instance, let us consider the case where

$\lambda \left(a\right)\propto {a}^{2}$. Then, during the slow-roll regime of the scalar field, for which

$V\left(\phi \right)\approx {\mathsf{\Lambda}}_{0}$, the effective value of the cosmological constant turns out to be discretized as

with,

${\mathsf{\Lambda}}_{k}^{\mathrm{eff}}\in ({\mathsf{\Lambda}}_{0},{\mathsf{\Lambda}}_{0}+\mathsf{\Lambda})$. If

${\mathsf{\Lambda}}_{0}\ll {M}_{P}^{4}$ and

$\mathsf{\Lambda}\sim {M}_{P}^{4}$, then, the interactions between universes might explain the apparent lack of potential energy for triggering inflation in the plateau like models that are enhanced by Planck [

10]. Despite the relative small value of the potential energy of the plateau with respect to the minima of the potential (see

Figure 2), the interactions among universes may excite the universe to a state of high value of

k. There, the absolute value of the potential would be high enough to trigger inflation in the universe, which may suffer afterwards a series of vacuum decays to reach a small value of the vacuum energy (see

Figure 2).

The second case that we can examine is the case where

$\lambda $ is a constant. Then, the Wheeler-DeWitt equation in the

${\tilde{\varphi}}_{k}$ representation turns out to be [

9]

with

and,

${E}_{k}={E}_{0}{sin}^{2}\frac{\pi k}{N}$. It effectively represents the quantum state of a universe with a radiation like content. Let us notice that in terms of the frequency

$\omega $, the effective value of the Friedmann equation is

where the last term is equivalent to a radiation like content for which the energy density goes like

${a}^{-4}$. For the flat branch the Friedmann Equation (

23) can analytically be solved yielding

where

${a}_{0}$ and

${\theta}_{0}$, are two constants of integration. The scale factor (

24) departures from the exponential expansion of a flat DeSitter spacetime at early times (see

Figure 3). This is a relevant feature because a radiation dominated pre-inflationary state in the evolution of the universe might have observable consequences in the properties of the CMB [

11] provided that inflation does not last for too long. It is remarkable then that some interacting processes in the multiverse might have observable consequences in the properties of a single universe like ours, although in this case it would not be distinguishable from an ordinary radiation content of the early universe.

Let us finally consider the case where the coupling function is proportional to

${a}^{-1}$. In that case the frequency is given, in the limit of a large number of universes

N, by

The last term of the frequency (

25) appears as well as a quantum correction to the Wheeler-DeWitt equation caused by the vacuum fluctuations of the wave function of the universe [

7]. It can be considered thus a sharp quantum effect having no classical analogue and thus a distinguishable effects of the interacting multiverse. The pre-inflationary stage induced in the evolution of the universe is more abrupt (a term

${a}^{-6}$ in the Friedmann equation) than those induced by a matter (

${a}^{-3}$) or a radiation (

${a}^{-4}$) content in the early universe. For the flat branch,

whose departure fro the exponential expansion of the DeSitter case is stronger than the other cases (see

Figure 4).