Does the quantum mechanics play any significant role in the contemporary cosmological dynamics? Could our universe as a whole behave as a quantum object? On a first glance, these questions make little sense: after all, the observable universe is an absolutely gigantic object, with the radius of the horizon estimated to be around \(10^{28}\ \hbox {cm}\). Instead, we are accustomed to encountering the quantum-mechanical phenomena on the atomic scales and below, and almost never – on the macroscopic scales and above. Of course, there are exceptions: the particular large-scale structure of the universe bears an indelible insignia of ancient quantum fluctuations. Still, even these fluctuations are only noticeable because they have been disproportionately blown up in the course of early cosmological inflation. In other words, the inflation in the young universe has served as a colossal amplifier of the quantum phenomena from the microscopic straight up to cosmological scales. But in the absence of anything comparable to the inflationFootnote 1 the universe at large appears to be essentially indifferent to the laws of quantum mechanics. Or is it?

One of the people who invented the idea that the universe as a whole can (and probably should) be treated as a quantum object was González-Díaz [2]. One of the authors (AYu) had a chance to work together with Pedro, and can attest to Pedro’s amazing capacity for generating very bold and novel ideas. AYu remembers one particular conversation, when PGD had first proposed the idea that the quantum effects might play a role on the cosmological scale. Initial reaction of AYu was admittedly rather sceptical: why would the quantum effects, negligible already on the scales of small bacteria (\(\approx 1 ~\upmu \hbox {m}\)) might suddenly show up on the scales of about 10 Gly (\(10^9\) light years)? PGD came up with a following answer: yes, the quantum effects are all but absent on the macroscopic scale. But this is simply a result of the decoherence phenomena, which suppresses the non-diagonal elements of the density matrix of a quantum system, and which manifests itself whenever that quantum system has any sort of interaction with the environment. But this mechanism does not work for the universe itself, for the universe has no outside environment, has nothing to interact with and is therefore immune to the effect of decoherence. Therefore, argued PGD, the universe cannot be treated as a classical object, because it can never cease to be a quantum one!

In this article we are going to develop the idea of quantum effects influencing the long-term cosmological dynamics, but we will go by a slightly different route. The key idea here would be not a decoherence (or a lack thereof), but instead a recent article by Hall et al. [1], which proposed a new and very novel way to formulate the quantum mechanics. After a little consideration, we have noticed that the method, proposed by Hall, Deckert and Wiseman allows for a very natural extension to the Friedmann cosmology, provided one indeed treats the universe as a quantum object, in full accord with the idea of PGD.

Once we started exploring this idea we have immediately noticed something very unexpected. It is always important to keep in mind that the cosmology, for all its seeming speculativeness, is still a branch of physics. Therefore, all of its ideas, no matter how crazy looking, have to permit for experimental (i.e. observational) verification. What we have noticed when we started working with our new idea is the following remarkable fact: in our model the quantum effects were inducing the dynamics identical to those produced by the hypothetical phantom fields. To be more precise, during a certain phase of its evolution the universe experiences the phantom acceleration (i.e. \(\ddot{H} >0\) whereas the parameter of state \(w<-1\)), which proves to be an entirely quantum phenomena.Footnote 2 The phantoms appeared to be a very unexpected but by no means unwelcome consequence of our proposed theory, so it makes sense to stop and discuss them in a little more detail.

If we are to create a list of the most popular topics in the contemporary cosmology, the term “phantom fields” would surely wind up somewhere on the top, being one of the most disputed and discussed subjects today. Taking, for example, all the articles published in ArXiv under the astro-ph rubric, and choosing among them the ones that contain the word “phantom” in the title, one will come out with more than 260 hits (and expanding this search to the hep-th rubric nearly doubles the number). If we then set our mind to studying these articles, we will soon realize that the majority of them follow a very specific guideline: taking a familiar cosmological model, adding to it a “phantom field”, solving the resulting equations (analytically if possible, numerically if not) and ruminating over the result. What is usually missing is a discussion of the physical nature of the proposed fields, which, taking into account the singularly mind-boggling properties these fields are supposed to possess (i.e. the wanton violation of the weak energy condition, the negative value of the kinetic term for the scalar field, etc), makes the entire enterprise seem rather dubious from a physical point of view. Of course, the usual retort to this objection would be a nod to the observational data as an undubitable proof of existence of a hidden type of matter suitably dubbed “dark energy” [6, 7]. We know it exists, we know it constitutes more than 68% of all matter in the observable universe and, finally, we also know that its parameter of state \(\omega \) might lie just below the vacuum energy threshold \(\omega =-1\) [8]. Hence, there is a solid chance that at least some of the “dark energy” is attributable to the “phantom fields” (i.e. fields with \(\omega < -1\)), so it is only natural to try out the various models that incorporate such fields. Sure, their real cause remains a mystery, but shouldn’t we at least try to guess their effect? While this mode of reasoning indeed has some optimistic charm, it unfortunately runs dangerously afoul of the normal scientific approach. We don’t mean to imply there is something inherently wicked and wrong with occasional forays into various possible models, done just to see what new effects and phenomena one might uncover; after all, the authors themselves had dabbled in their fair share of unusual cosmologies (such as the universes with the varying speed of light [9] and brane worlds [10]), and even did some treatises on various aspects of the phantom cosmologies (from the universe-encompassing phenomena of the “Big Trip” [11] to the problem of proliferation of Boltzmann brains within the universe filled with the phantom fields [12]). But to have almost the entire field of research succumb to this “throw it at the wall and see what sticks” approach is truly utterly frustrating. Because, to put it bluntly, the science simply does not work this way. In fact, if one is is to put an analogy with a cosmological breakthrough of a similar calibre, a Penzias and Wilson’s discovery of the cosmic microwave radiation (CMR) [13] comes to mind. But in order to make the analogy complete imagine this discovery accompanied not by the corresponding article by Dicke et al. [14], or any other work providing the explanation of the phenomena by drawing the relationship to the already familiar Big Bang theory. No, instead picture the cosmologists viewing the apparently ubiquitous background radiation as a byproduct of equally ubiquitous and ever-present “luminous” matter, and then dedicating article after article to an ambitious tusk of construction of an apt model of the universe filled with such a thing. Imagine, how much time and energy could have been wasted in such a pursuit... Similarly, the understanding we now have of the later stages of stellar evolution in many ways stems from the fact that the research team that has first detected the pulsar signature [15] chose to look for its explanation in the predictions of the fundamental theoretical physics (i.e. in the behaviour of strong magnetic fields of rapidly rotating neutron stars), and not in the realm of the hypothetical extraterrestrial interstellar communications (although the team actually gave the signal a codename “LGM-1”, which of course stands for “little green men”).

To put it in other words, what we have now is a proliferation of the articles that attempt to explain what the phantom fields do, a drought of those that explain what these fields are – and practically none that have a benefit of showing where these fields would fit in in the known laws of nature.Footnote 3 So, we have to ask ourselves: is there any way out of this self-imposed predicament? If the history of science (e.g., the aforementioned discovery of CMR) is any indication, the solution to the conundrum of the “phantom fields” should be looked for at the most fundamental laws of nature. How deep shall we look? Apparently, the judge of that would be the very properties of these fields. Since they are so radically different from literally every other type of matter (be it dust, radiation or even a vacuum energy), it is a safe assumption that to understand their physical nature we should start at the vary basic level – where a quantum mechanical laws reign supreme. And it may be that we already have all the necessary ingredients for that.

Enter the 2014’s article [1] by Hall et al. In this article the authors have presented a very novel outlook on the problem of a deterministic interpretation of the quantum mechanics. Two such interpretations were known. First of all, there was the de Broglie-Bohm wave-pilot interpretation (dBB) [16], which describes the behaviour of the quantum particles as the one that obeys the classical Newtonian mechanics with a “twist”: the equation of motion includes a contribution of an additional field – a “quantum potential”, which depends on the absolute value of the Schrödinger’s wave function, and is wholly responsible for all the marvelous quantum-mechanical idiosyncracies. Secondly, there was the famous many-worlds interpretation of Hugh Everett III [17], postulating that all the outcomes (no matter how improbable) of any given quantum-mechanical observation are actually physically realized – each in its own separate “universe” peacefully coexisting in a broader multiverse; the very act of observation simply slips us into one of those universes. Both of those interpretations have their own benefits; but both although have their own problems. For example, dBB has to postulate the existence of a certain field that directs and controls the movement of a given quantum particle, but is not directly produced by it – or any other particle for that matter. The many-worlds interpretation in its turn had to work with and predict the existence of uncountably many “quantum worlds” (to fill up the entire Hilbert space), including very bizarre ones. Furthermore, the worlds themselves do not exert any influence on each other but apparently exist for the sole purpose of accepting the stray visitors after they conducted their “observations” – like a chain of free motels at a sports event...

Now, the crux of the matter is that both of these interpretation has to produce one the same quantum mechanics we know. Therefore, there should theoretically exist a way to combine these two seemingly incompatible world-views. In essence, that is exactly what the authors of [1] did. They started out with the many-worlds prediction of an existence of multiple classical worlds. First thing they did was to radically cut down the number of these worlds to just \(N \in {\mathbb {Z}}\) (so there should be at most countably many of them). Furthermore, these classical worlds should not just co-exist with each other, but should in fact interact, the mechanism of interaction being the de Broglie-Bohm mechanism; in fact, the limit \(N \rightarrow \infty \) should produce exactly the same potential as in the dBB framework, except this time it will not be a by-product of some arcane “external” field, but a result of the repulsive forces between the neighbouring classical worlds. So, we end up with a picture that successfully combines the two seemingly incompatible interpretations and at the same time actively avoids the weaknesses of both of its predecessors – it literally takes the best of both worlds – or, in this case, “the best of both multiverses”.

We will talk more about this interpretation (known as the interpretation of many interacting worlds a.k.a. MIW) and how it can derived using the dBB formalism in Sect. 2, but right now we are interested in just two of its most important predictions. First of all, MIW predicts that the force of interaction between the different “worlds” is repulsive and that it decreases monotonously with the relative distance between them. This implies that the reason the quantum phenomena are so fragile has nothing to do with a “collapse of a wave function” (whatever that means) – in fact, such an object as a wave function is inessential and can be completely avoided in the MIW formalism. No, the existence of quantum phenomena relies solely on the mutual positions of the neighbouring “worlds” – when they are sufficiently close, the quantum potential is alive and kicking; when they depart, the quantum potential abates and the particles become effectively classical again. There is no such thing as “decoherence”. On the other hand, the similar arguments can be applied to not just one observable particle (and its corresponding “world”, aptly called in [1] the “world-particle”), but also to the progressively larger groups and structures made of said particles; if the ideology of MIW is correct and there is no decoherence, such macroscopic objects should also be responsive to the quantum-mechanical effects, produced by their quantum “shadow copies”, provided, that is, that the positions of the particles in the “doppelgänger worlds” are sufficiently close to those in the observable system.

Naturally, for an ordinary macroscopic object the force exerted by this type of interacting is expected to be vary small. Indeed, if we estimate the probability that the majority of atoms in a human body gets sufficiently close to its its “doppelgängers”, it would be inversely proportional to the whole number of such atoms and so be roughly equal to \(10^{-27}\). This, of course, explains why the quantum mechanics and its phenomena are mostly relegated to the domain of very small (atomic-sized) objects. However, unlike the macroscopic objects (e.g., protists, potatoes, people, planets etc.), on the cosmological scale of entire Friedmann universe, the impediment disappears, owing to the fact that the cosmological dynamics of a Friedmann universe is completely equivalent (with one minor assumption regarding the spectrum of possible energies – see Sect. 3) to a dynamics of a single material point situated on a surface of a homogeneous and isotropic gravitating sphere.

Here we arrive at a second important consequence of MIW: the quantum objects strictly obey the Newtonian physics, just with an additional force induced by the world-to-world interactions. In fact, from the point of view of the Hamiltonian formalism, the quantum physics merely requires an addition of one special term to the Hamiltonian. But it is important to remember that the Einstein–Friedman equations that govern the evolution of the homogeneous isotropic universe can also be derived from the Newton equations describing the adiabatic expansion of a homogeneous isotropic 3-dimensional sphere of matter – the only contribution required by the General Relativity and absent from the Newtonian physics being the limitation on the possible spectrum of the curvature constant \(\kappa \) (in fact, \(\kappa ={-1,0,1}\), which corresponds to the open, flat and closed universes). It is also a rather straightforward job to write down a corresponding Hamiltonian (which, of course, corresponds to the one, produced via the General Relativity formalism). If our understanding of the quantum mechanics via MIW is indeed correct, then in addition to the observable universe there should exist a multitude (perhaps even countably many) of shadow universes, that can interact with ours via a quantum potential when these shadow universe are “close enough” – but this time instead of the relative positions of the universes (and yes, this time we are indeed talking not about the “worlds”, but “universes”, and one can not really define a “position of a universe”) – we have to compare their “states”, which geometrically can only mean measuring how similar (or dissimilar) their sizes are, i.e. we have to measure their relative scale factors. After all, it is the scale factor that replaces the single coordinate in the Einstein–Friedmann equations written in the Hamiltonian formalism, so it makes sense to use it in the MIW extension for cosmology. However, one has to be very careful when implementing this idea. While a construction of the corresponding equations (that effectively combine the Einstein–Fridman equations with the quantum interaction) is straightforward enough (as will be demonstrated in Sect. 3), it is some of the properties of their special solutions that that have to be pondered. For the sake of simplicity, let us consider a special case of just two interacting universes. If we don’t impose any sort of initial conditions on their dynamics, it is not impossible to obtain a special solution for the scale factors of these two universes with one of them turning to zero at some moment of time \(t_0\) while the other one stays strictly positive both at and in an open neighbourhood of \(t_0\) – in fact, in Sect. 3 we show that such solutions indeed exist (for the special case of two empty universes, filled solely with the quantum potential). Now this presents a problem. The scale factor turning to zero means that at \(t=t_0\) the first universe collapses into a singularity. Hence, at a close vicinity of \(t_0\) there should be a brief time period when the entire collapsing (for \(t<t_0\)) universe is microscopic in size. Such a universe should be under a tremendous influence of the quantum effects. However, we remind the reader that we are working within the framework of MIW paradigm, which explicitly defines all quantum phenomena as manifestations of the repulsive interaction between the neighboring universes – in our case, the interaction between the first and the second universes. The problem then is that this repulsion can only be significant when the relative scale factors are close to each other, that is, both of them has to be equal to zero at \(t=t_0\) – and this is definitely not our case.

The only sensible way to eliminate this problem would be to assume that all scale factors of all available interacting universes should hit zero at exactly the same time, that is – they all have to be proportional to each other. In other words, all these scale factors shall be proportional to one, special scale factor which we have called the master-factor. This simple deduction (dubbed the “master-factor method”) not only resolves the singularity problem we just discussed, but also greatly simplifies the resulting equations, and, most surprising of all, provides a very new outlook on the multiverse itself, asserting that the properties of the observable universe in a strange way depend on a point of view of the observer. If, for example, one was to look at the multiverse from a bird’s eye perspective (assuming, for the sake of the argument, that a bird can actually hover over and look upon the entire multiverse), one was to see but one universe, its size determined by the “master factor” and its evolution determined by just one ordinary differential equation. However, this is not what the individual observers inhabiting their respective universes will see. These “down-to-earth” (or, actually, “down-to-a-universe”) observers, would have their own view (which can be called a “frog’s” perspective to distinguish it from the “bird’s” perspective described above [18]), with the effective observable scale factor, density, pressure and curvature being potentially quite distinct from those observed by both the “bird” and the other “frogs” (belonging to the different universes), proving that in some way not only beauty, but the universe itself lies in the eyes of the beholder.

Thus, equipped with both the new modified Einstein–Friedmann equations and the master-factor method, in Sect. 4 we embark on exploration of their possible consequences for the fate of the observable universe. In particular, we demonstrate that even for the simplest case of just two interacting universes, filled with ordinary matter (\(\omega =0\)), radiation (\(\omega =1/3\)) and the vacuum energy (\(\omega =-1\)), by adding the quantum interaction it becomes possible to replicate the effects akin to those of a dark matter and the phantom fields (\(\omega < -1\)); it might even act to prevent a complete collapse of the universe to a final singularity (Sect. 5), thus eliminating the very point where our knowledge of physics seems to officially break down.

But before we move on to these prospects, it would be a good idea to take a look a the theories that gave rise to them, namely: the De Broglie–Bohm and the Many Interacting Worlds interpretations.