# What if? Exploring the multiverse through Euclidean wormholes

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## Abstract

We present Euclidean wormhole solutions describing *possible bridges* within the multiverse. The study is carried out in the framework of third quantisation. The matter content is modelled through a scalar field which supports the existence of a whole collection of universes. The instanton solutions describe Euclidean solutions that connect baby universes with asymptotically de Sitter universes. We compute the tunnelling probability of these processes. Considering the current bounds on the energy scale of inflation and assuming that all the baby universes are nucleated with the same probability, we draw some conclusions about which universes are more likely to tunnel and therefore undergo a standard inflationary era.

## 1 Introduction

Humankind has, ever since history can tell, been looking for possible answers and hints to the questions: Where do we come from? Where are we heading to? Cosmology is the path to address these questions on scientific grounds. As regards the first question, general relativity predicts the existence of a past big bang, at least for standard matter and for a homogeneous and isotropic universe, a singularity which is hoped to be wiped out through a primordial quantum era [1]. At the semi-classical level, this may imply the existence of Euclidean solutions or instantons that geometrically describe Euclidean wormholes or bridges in spacetime, in its widest sense, where the big bang singularity is circumvented or at least shadowed by the presence of Euclidean wormholes [2, 3, 4, 5, 6, 7] connecting baby universes to some Lorentzian singularity-free universes.

While Euclidean wormholes can be seen as a natural geometrical and gravitational extension of the Coleman–De Luccia instanton [8], where the latter assumes the existence of two vacua and a decaying process connecting them, the former do not necessarily assume the existence of such vacua. In both cases, however, an extremisation of the Euclidean action is reached. In addition, Euclidean wormholes are not exclusive to general relativity and can be found, for example, in string theory (see, for example, [9, 10]). Moreover, even though in general one uses the term “Euclidean wormhole” to describe a curved Euclidean space with – at least – two large asymptotic regions connected by a throat, this idea has been extended as well for spacetimes with a cosmological constant that can be positive [6, 11] or negative [5, 12, 13]. We would like to highlight that, strictly speaking, a Euclidean wormhole – in the simplest geometrical setup – is constructed from two glued instantons [14, 15]. However, given that in both cases Euclidean wormholes and instantons are Euclidean bridges connecting different Lorentzian spacetimes, we will use the words “instanton” and “wormhole” indistinguishably.

It is therefore natural to assume the existence of wormholes as *connecting paths* not only within remote regions of the universe but also within the multiverse as a whole [16]. In this regard, we would like to stress that the paradigm of inflation, supported observationally since COBE’s first measurements [17] of the Cosmic Microwave Background anisotropies and the subsequent experiments WMAP [18] and Planck [19], predicts and supports the existence of the multiverse. In fact, the multiverse concept can be reached and understood from several approaches, from the seminal idea of Everett of a multiverse formed by the branches of quantum mechanics [20], to the landscape of the string theories [21], the inflationary multiverse [22, 23], or the ekpyrotic scenario [24, 25], among many others [26, 27, 28, 29, 30, 31, 32, 33, 34]. In any case, one assumes the existence of an undetermined number of realisations of the universe, each one causally separated from the others by the presence of quantum barriers, event horizons or extra dimensions. Yet, their quantum states may still be related by the existence of non-local correlations in the global quantum state of the spacetime and the matter fields.

In this paper, we show the existence of wormhole solutions in the framework of third quantisation, one of the current proposals to describe the multiverse. It basically consists of considering the solution of the Wheeler–DeWitt equation as a field that propagates in the minisuperspace of spacetime metrics and matter fields, and thus quantising the wave function of the spacetime and matter fields by following a formal parallelism with the customary procedure of a quantum field theory (see Refs. [35, 36, 37, 38]). Then the creation and annihilation operators of the third-quantisation formalism describe the creation (or the annihilation) of a particular spacetime–matter configuration. In particular, the solutions that we have found correspond to *Euclidean tunnels* that connect baby universes with asymptotically de Sitter universes. In this regard, we have focussed exclusively on the tunnelling between two Lorentzian universes supported by the same scalar field with the same initial kinetic energy. A more complete analysis would consider the “communication” or tunnelling between universes with different initial kinetic energies for the scalar field. A further possibility to describe the “communication” between different universes of the multiverse is by considering entangled universes [39, 40].

The paper can be outlined as follows. In Sect. 2, we summarise the model we will be analysing and review briefly the third-quantisation approach. Then we find for the first time exact solutions describing an instanton within this framework in the presence of a minimally coupled massive scalar field. In this framework and within a semi-classical approach, the multiverse can be seen as a collection of semi-classical universes with a label that indicates the initial kinetic energy of the scalar field that supports them. In Sect. 3, for a given universe within the multiverse; i.e. a given label, we obtain the transition probability describing the tunnelling from a baby universe to an asymptotically de Sitter universe. Our calculations assume the tunnelling boundary conditions of Vilenkin [41, 42]. Finally, in Sect. 4, we present our conclusions. For clarity, we include as well an appendix where we obtain analytically the transition amplitude analysed in Sect. 3.

## 2 Model

*a*containing a minimally coupled scalar field \(\varphi \) with mass

*m*and quadratic potential \(\mathcal {V}(\varphi ) = \frac{1}{2}\,m^2\varphi ^2\). The Wheeler–DeWitt (WDW) equation for the wave function \(\phi (a,\varphi )\) of such a configuration of the spacetime and matter field

^{1}reads [44]

*a*, but such a term would not influence our calculations.

*K*are related to the momentum conjugated to the scalar field, \(p_\varphi \), and we interpret the decomposition (2.6) in the way that each amplitude \(\phi _K(a)\) of the wave function \(\phi (a,\varphi )\) represents a single universe with a specific value of \(p_\varphi \). The wave functions of the universes satisfy then the following effective WDW equation:

^{2}

*K*appearing in the last expression is defined as

*t*via \(\mathrm {d}\eta /\mathrm {d}t = a^{-1}\), we need to solve the following differential equation:

*K*for which the tunnelling effect happens. In the remaining part of this section, we present the solutions of Eq. (2.20) in the Lorentzian regions \(0<x<x_-\) and \(x_+<x\) and in the Euclidean region \(x_-<x<x_+\) for the non-trivial cases \(\alpha _K\in (0,\,\pi )\).

### 2.1 Baby universe: \(0< x < x_-\)

^{3}(cf. Eq. 17.4.63 in Ref. [48]):

*x*from a maximum value \(\xi _-(x=0)= \arccos [(\sin [(\pi -\alpha _K)/6]/\cos [\alpha _K/6])^{1/2}]\) to \(\xi _-(x=x_-)=0\). Upon substitution in Eq. (2.20) and after an integration from \(\eta \) to \(\eta _-:=\eta (x_-)\), we find that the conformal time fulfills

### 2.2 Asymptotically de Sitter universe: \(x_+< x < +\infty \)

### 2.3 Euclidean wormhole: \(x_-< x < x_+\)

^{4}(cf. Eq. 17.4.69 in Ref. [48]):

*x*from \(\tilde{\xi }(x=x_-)=\pi /2\) to \(\tilde{\xi }(x=x_+)=0\). When replacing (2.28) in (2.20) and integrating in Euclidean time \(\tilde{\eta }=\mathrm {i}\eta \) from \(\tilde{\eta }\) to \(\tilde{\eta }_+=\tilde{\eta }(x=x_+)\) we find that the conformal Euclidean time fulfills

In Fig. 2 we depict the combined evolution of the squared scale factor during the two Lorentzian regions and through the Euclidean wormhole. During the baby universe phase (depicted in red), the scale factor evolves from 0 to \(a_-\) as the time displacement \(\Delta \eta :=(\eta -\eta _-)/|\eta _{(a=0)}-\eta _-|\) varies from \(-1\) to 0. As the scale factor reaches the value \(a_-\), the universe can enter a Euclidean wormhole (depicted in blue), in which the scale factor grows from \(a_-\) to the maximum value \(a_+\) as the Euclidean time displacement \(\Delta \tilde{\eta }:=(\tilde{\eta }- \tilde{\eta }_+)/|\tilde{\eta }_{(a=a_-)}-\tilde{\eta }_+|\) goes from \(-1\) to 0. Once the value \(a_+\) is reached, the universe exits the Euclidean wormhole and enters a near de Sitter expansion (depicted in green). In this final phase the scale factor grows in an accelerated fashion as the time displacement \(\Delta \eta :=(\eta -\eta _+)/|\eta _{(a=+\infty )}-\eta _+|\) varies from 0 to 1.

## 3 Tunnelling

*K*, we need to evaluate the following integral:

*x*, \(x_+\), \(x_-\) and \(x_0\) are defined in (2.19). The integral (3.2) becomes trivial for the special case of \(K=0\), the creation of an expanding universe from nothing, as computed in [41]. In this case we find \(I_{(K=0)} = (1/3)x_+^{3/2}\).

*I*can be solved by means of the transformation

*I*as a linear combination of the complete elliptic integrals of the first, second and third kind [48],

*K*(

*m*),

*E*(

*m*), \(\Pi (n|m)\), respectively:

*I*is presented in Appendix A 2.

*K*of the baby universe is to the maximum quantum allowed value. In Fig. 3 we plot the tunnelling probability as a function of \(\gamma \) and \(K/K_\text {max}\), while in Fig. 4 we present the tunnelling probability as a function of \(\gamma \) for different ratios \(K/K_\text {max}\) and as a function of \(K/K_\text {max}\) for different values of \(\gamma \). One can see that the tunnelling probability goes to 1 for \(K \rightarrow K_\text {max}\), as expected, since the Euclidean region ceases to be present in that limit. For \(K\approx 0\) the tunnelling probability approaches the solution for the creation of an expanding universe from nothing [41], which is marked by a blue dashed line. Due to the \(1/\gamma \) factor in the argument of the exponential, the tunnelling probability decays rapidly for low values of \(\gamma \). As such, if the scale of inflation is well below the Planck scale, then the tunnelling probability is extremely low except for \(K\approx K_\text {max}\), as can be observed in Fig. 3.

## 4 Conclusions

Within the framework of the third quantisation, one of the current proposals to describe the multiverse, we have shown the existence of Euclidean wormhole solutions which describe possible bridges within the multiverse.

More precisely, by considering a massive minimally coupled scalar field within the framework of the third quantisation, we can describe a whole bunch of universes fingerprinted by the initial kinetic energy of the scalar field that supports them. It turns out that for a given initial kinetic energy of the scalar field two classically disconnected solutions emerge: a baby universe and an asymptotically de Sitter universe. Although these two Lorentzian solutions are classically disconnected, it turns out that this is no longer the case from a semi-classical point of view. In fact, as we have shown, these two solutions are connected through a Euclidean wormhole (cf. Fig. 2 for a schematic representation). In addition, our solution generalises the Giddings–Strominger instanton [4], even though the two solutions have a completely different origin. While our solution is constructed in the framework of the third quantisation with a massive minimally coupled scalar field, the Giddings–Strominger instanton is constructed in the framework of string theory and is supported by an axion whose field strength tensor is defined through a rank-three anti-symmetric tensor.

Assuming the transition amplitude between two Lorentzian universes proposed in [41, 42], we have calculated the probability of tunnelling from the baby universe to the asymptotically de Sitter universe. Our results are graphically represented in Figs. 3 and 4. We can conclude that the larger the initial kinetic energy of the scalar field is, the higher is the probability of the baby universe to cross the barrier depicted in Fig. 1; i.e. the higher is the probability that the universe crosses towards the inflationary era through the shortcut provided by the Euclidean wormhole.

Finally given that the highest value of the potential barrier (see Fig. 1) separating the two Lorentzian universes is related to the scale of inflation, i.e. \(H_\mathrm {dS}\) (cf. for example Eq. (2.8)), of the asymptotically de Sitter universe, we can estimate \(K_{\max }\) defined in Eq. (2.15) or equivalently the parameter \(\gamma \) given in the same equation. In fact, given that the energy scale of inflation is at most of the order \(8.8\times 10^{19}\) GeV [51], we can conclude that \(\gamma \) must be smaller than \(5.2029\times 10^{-11}\) or equivalently \(K_{\max }\) must be quite large. Therefore, what we have proven is that if all the baby universes are nucleated with the same probability, those with larger *K* are most likely to tunnel through the wormhole and therefore undergo an inflationary era like our own patch of the universe. In a subsequent paper, we will constrain the current model with CMB data following our previous work [52, 53].

## Footnotes

- 1.
The wave function \(\phi (a,\varphi )\) represents the quantum state of the spacetime and the matter fields all together. Hence, the traditional name

*wave function of the universe*[43]. However, this name can be misleading in the multiverse scenario we are working on, so we shall just call it the wave function of the spacetime and matter fields. - 2.
A stiff matter content of the universe would introduce a similar term, proportional to \(a^{-6}\), in the Friedmann equation. However, the term here is also proportional to \(\hbar \), which reveals its quantum nature without classical analogue (\(\hbar \rightarrow 0\)).

- 3.
We point out that this transformation is not valid for \(\alpha _K=0\), since in that case \(x_0=x_-=0\) and the argument inside the \(\arccos \) on the right-hand side of Eq. (2.21) diverges. Physically this can be understood by the fact that for \(\alpha _K=0\) there is no baby universe – the expanding asymptotically de Sitter universe is created from nothing [41].

- 4.
We point out that this transformation is not valid for \(\alpha _K=\pi \), since in that case \(x_+=x_-\) and the argument inside the \(\arccos \) on the right-hand side of Eq. (2.28) diverges. Physically this can be understood by the fact that for \(\alpha _K=\pi \) there is no Euclidean region.

## Notes

### Acknowledgements

This article is based upon work from COST Action CA15117 “Cosmology and Astrophysics Network for Theoretical Advances and Training Actions (CANTATA)”, supported by COST (European Cooperation in Science and Technology). The work of M.B.-L. is supported by the Basque Foundation of Science Ikerbasque. She and J.M. also wish to acknowledge the partial support from the Basque government Grant No. IT956-16 (Spain) and FONDOS FEDER under grant FIS2014-57956-P (Spanish government). The research of M.K. was financed by the Polish National Science Center Grant DEC-2012/06/A/ST2/00395 and by a grant for the Short Term Scientific Mission (STSM) “Multiverse impact onto the cosmic microwave background and its relation to modified gravity” (COST-STSM-CA15117-36137) awarded by the above-mentioned COST Action. M.K. and J.M. would like to thank the Centro de Matemática e Aplicações of the Universidade da Beira Interior in Covilhã, Portugal for kind hospitality while part of this work was done. M.K. also thanks M.P. Dąbrowski for fruitful discussions. J.M. is thankful to UPV/EHU for a Ph.D. fellowship.

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