 the cosmological constant can be affected by the interaction among the universes.

This cosmological picture is then completely different than the one single universe picture. Interactions and collective behaviour might then occur among the universes of the multiverse. Actually, this collective behaviour is fairly general and is at the very heart of quantum theory, which is a non-local theory and within which all the physical elements are fundamentally coupled to their environment and individual properties arise out of a result of some decoherence process. Thus, the true quantum state of the space-time must account for the states of all the universes, if they exist. The aim of this paper is to examine whether some of these collective processes may have an observable influence on the properties of our universe.

Irrespective of the consideration of a multiverse and its implications, it seems therefore interesting to analyze the influence, if any, that different distant regions of the space-time may have on the properties of the observable part of our local universe (see also Ref. \cite{Alonso-Serrano:2014dsa}) with a two-fold aim: i) to analyze whether they might help to solve some of the open questions posed by the latest Planck data \cite{Ade2014, Ijjas2013}, and ii) to look for distinguishable imprints of other universes in, for instance, the properties of the cosmic microwave brackground (CMB) spectrum \cite{Aguirre:2007an}.

This paper is organized as follows: In section II, we discuss the Hamiltonian quantum cosmology model of an interacting multiverse. In section III, we consider the bubble formation, that is, the nucleation of universes in a parent space-time and specifically address this nucleation in a setting where the universes are interacting. Finally, in section IV, we present a discussion of our results.



\section{The interacting multiverse}


Let us consider a simply connected piece of a homogeneous and isotropic space-time manifold endowed with a scalar field $\varphi$ that represents the matter content. More general topologies can also be considered by splitting the whole manifold into simply connected pieces of space-time \cite{Hawking1988}, each of which is quantum mechanically described by a wave function $\phi = \phi(a,\varphi)$ that is the solution of the Wheeler-DeWitt equation \cite{Alonso2012}
\begin{equation}\label{WDW01}
\ddot{\phi} + \frac{1}{a} \dot{\phi} - \frac{1}{a^2} \phi'' + \omega^2(a,\varphi) \phi = 0,
\end{equation}
where the scalar field has been rescaled according to Ref. \cite{Garay2013}, $\varphi \rightarrow \frac{2}{M_P}\sqrt{\frac{\pi}{3}} \varphi$, where $M_P$ is the Planck mass. In Eq. (\ref{WDW01}) the dots represent derivatives with respect to the scale factor and the prime denotes derivative with respect to the scalar field. The function $\omega(a,\varphi)$ contains the potential terms of the Wheeler-De Witt equation. In the case of a closed  space-time it is given by
\begin{equation}\label{FRQ01}
 \omega^2(a,\varphi) \equiv \sigma^2 (H^2 a^4 - a^2 ),
\end{equation}
where $\sigma \equiv \frac{3 \pi M_P^2}{2}$ and $H\equiv H(\varphi)$ is the Hubble function. The frequency $\omega$ has units of mass or, equivalently, units of the inverse of time or length. We shall consider two contributions to the Hubble function, i.e., $H^2 = H_0^2 + H_1^2$. The first one is due to the existence of a cosmological constant, $H_0^2={\Lambda_0 \over {3 M_P^2}}$, which is assumed to be very small. The second contribution is due to the potential of the scalar field, $H_1^2 = \frac{8 \pi}{3 M_P^2} V(\varphi)$.

Let us now develop a quantum field theory for the wave function $\phi$ in the curved minisuperspace spanned by $(a, \varphi)$ with a minisuperspace metric given by
\begin{equation}\label{MSM}
 G_{M N} = \left( \begin{array}{cc}
-a & 0  \\
0 & a^3 \end{array} \right),
\end{equation}
where $M, N$ stands for $\{a,\varphi\}$. The \emph{line} element of the minisuperspace metric is therefore
\begin{equation}\label{LE}
  d\mathfrak{s}^2 = - a d a^2 + a^3 d\varphi^2.
\end{equation}
The scale factor, $a$, formally plays the role of the time variable  and the matter field the role of the spatial variable in the two dimensional Lorentzian minisuperspace metric (\ref{MSM})  ($a(t)$ can actually be seen as  a time reparametrization). We can now follow the usual procedure of a quantum field theory for the scalar field\footnote{Spinorial and vector fields could also been considered.} $\phi(a,\varphi)$ by considering the following action
\begin{equation}\label{S01}
 S = \int da d\varphi \, \, \mathcal{L}(\phi, \dot{\phi}, \phi'),
\end{equation}
where the Lagrangian density is given, as usual, by
\begin{eqnarray}\label{L01}
  \mathcal{L} &=& \frac{1}{2} \sqrt{-G} \left\{ G^{M N} \partial_M \phi \partial_N \phi - \mathcal{V}(\phi)\right\} \\
   &=& \frac{1}{2} \left( - a \dot{\phi}^2 + \frac{1}{a} \phi'^2 \right) + \frac{a \, \omega^2}{2}  \phi^2,
\end{eqnarray}
where $G = {\rm det}(G_{M N})$. Then, the corresponding Euler-Lagrange equation \cite{Birrell1982}
\begin{equation}\label{EL01}
\frac{1}{\sqrt{-G}} \partial_M\left( \sqrt{-G} G^{M N} \partial_N \right) \phi + \frac{1}{2} \frac{\delta \mathcal{V}}{\delta \phi} = 0,
\end{equation}
turns out to be the Wheeler-De Witt equation, Eq. (\ref{WDW01}).

The Hamiltonian density that corresponds to the Lagrangian density, Eq. (\ref{L01}), is given by
\begin{equation}\label{H01}
 \mathcal{H} = -\frac{1}{2} \left( \frac{1}{a} P_\phi^2 + \frac{1}{a} \phi'^2 + a \omega^2 \phi^2 \right),
\end{equation}
where
\begin{equation}\label{MO01}
  P_\phi \equiv \frac{\delta \mathcal{L}}{\delta \dot{\phi}} = - a \dot{\phi}~,
\end{equation}
is the momentum conjugated to the field $\phi$.

We can now  pose an interaction scheme \cite{Bertolami2008, Alonso2012} among a set of $N$ universes by considering a total Hamiltonian density given by \cite{Alonso2012}
\begin{equation}\label{H02}
  \mathcal{H} = \sum_{n=1}^{N} \mathcal{H}_n^{(0)} + \mathcal{H}_n^{(i)},
\end{equation}
where $\mathcal{H}_n^{(0)}$ is the unperturbed Hamiltonian density of the $n$-universe, given by Eq. (\ref{H01}), and $\mathcal{H}_n^{(i)}$ is the Hamiltonian density of the interaction for the $n$-universe, that here we consider as the simple quadratic interaction between next neighbour universes,
\begin{equation}\label{H03}
 \mathcal{H}_n^{(i)} =  \frac{a \lambda^2(a)}{8} (\phi_{n+1} - \phi_n)^2,
\end{equation}
where $\lambda(a)$ is a coupling function that can depend on the value of the scale factor and we use periodic boundary conditions so that, $\phi_{N+1} \equiv \phi_1$.

We consider that the Hamiltonian density, Eq. (\ref{H02}), represents the evolution of a set of universes that are interacting to each other, where each internal observer do not see any interaction but only its own Hamiltonian density. We can take into account,  for the sake of simplicity, a new representation given in terms of the normal modes by means of the Fourier transformation of $\phi$ and $P_\phi$
\begin{eqnarray}
  \tilde{\phi}_k &=& \frac{1}{\sqrt{N}} \sum_{n=1}^{N} e^{-(2\pi i k n/N)} \phi