Nonlocal Quantum Effects in Cosmology

Since it is commonly believed that the observed large-scale structure of the Universe is an imprint of quantum fluctuations existing at the very early stage of its evolution, it is reasonable to pose the question: Do the effects of quantum nonlocality, which are well established now by the laboratory studies, manifest themselves also in the early Universe? We try to answer this question by utilizing the results of a few experiments, namely, with the superconducting multi-Josephson-junction loops and the ultracold gases in periodic potentials. Employing a close analogy between the above-mentioned setups and the simplest one-dimensional Friedmann-Robertson-Walker cosmological model, we show that the specific nonlocal correlations revealed in the laboratory studies might be of considerable importance also in treating the strongly-nonequilibrium phase transitions of Higgs fields in the early Universe. Particularly, they should substantially reduce the number of topological defects (e.g., domain walls) expected due to independent establishment of the new phases in the remote spatial regions. This gives us a hint for resolving a long-standing problem of the excessive concentration of topological defects, inconsistent with observational constraints. The same effect may be also relevant to the recent problem of the anomalous behavior of cosmic microwave background fluctuations at large angular scales.


Introduction
The concept of quantum nonlocality originates actually from the paper by Einstein, Podolsky, and Rosen (EPR) [1], who posed the problem of correlation between the measurements of two physical objects located in the causally-disconnected regions of space, i.e., beyond the mutual light cones. In the modern and most frequently used in the experiments form, this phenomenon can be illustrated in Figure 1. Here, an original particle of zero spin decays at the instant t = 0 into two particles with equal but oppositely-directed spins s 1 and lieved now that the observed large-scale structure of the Universe is an imprint of quantum fluctuations existing at the very early stages of cosmological evolution. Therefore, it becomes interesting to pose the question: Can the effects of quantum nonlocality, similar to EPR correlations, manifest themselves also in cosmology? To answer this question, it should be kept in mind that Higgs field, filling the entire Universe and giving masses to the elementary particles, represents actually a kind of Bose-Einstein condensate (BEC), i.e., a macroscopic quantum state, in which the specific quantum correlations should naturally occur. Besides, it is well known that the most important feature in temporal dynamics of the Higgs field is a phase transition caused by an evolving temperature of the Universe [2], which can finally result in the formation of nontrivial states of the physical vacuum.
In fact, the problem of "complex" vacuum was recognized long time ago, just after appearance of the idea of spontaneous symmetry breaking in the quantum field theory. Particularly, in the Conference on the occasion of the 400th anniversary of Galileo Galilei's birth, held in Pisa in 1964, Bogoliubov emphasized that "it is hard to admit, for example, that the 'phases' are the same everywhere in the space. So it appears necessary to consider such things as 'domain structure' of the vacuum" [3]. In the next decade, the problem of formation of the domain walls in the course of cosmological evolution was considered in much detail in the work by Zeldovich, Kobzarev, and Okun [4] and later in papers by many other authors.
(A quite comprehensive overview of the domain wall dynamics was given, for example, in paper [5].) A commonly-accepted scenario of formation of the domain walls by a phase transition in the early Universe is illustrated in Figure 2. A uniform initial ("symmetric") state of the Higgs field, existing soon after the Big Bang, cools down due to expansion of the Universe; so that the symmetric phase of the field becomes energetically unfavorable, and the seeds of a new ("symmetry-broken") phase emerge in some spatial points. (For the sake of simplicity, we shall assume that these points are equally separated along the coordinate r and originate at the same instant of time t = 0.) Since the seeds of the low-temperature phase emerge independently in the remote spatial points, their states of the degenerate vacuum, in general, will be different from each other.
Then, the domains of the new phase quickly grow in the course of time and at the instant t = t 0 collide with each other. However, if the values of the symmetry-broken phase in two neighboring domains were different from each other, they cannot merge smoothly. Instead, a stable topological defect-domain wall (or "kink")-should be formed at their boundary. So, if the initial separation between the seeds of new phase was ξ, then the resulting concentration of the domain walls can be roughly estimated as n ≈ 1/ξ. This is the so-called Kibble-Zurek (KZ) mechanism for the formation of topological defects after the strongly-nonequilibrium phase transformations [6,7].
Strictly speaking, scenario outlined in Figure 2 refers to the simplest case of Higgs field, possessing Z 2 symmetry group (i.e., admitting a discrete symmetry breaking). However, the same basic idea is applicable also to more realistic Higgs fields with the continuous symmetries, whose breaking can lead to the formation of more complex defects of the vacuum, such as the monopoles and cosmic strings (or vortices). In general, KZ mechanism gives the following estimate for concentration of the topological defects: where ξ eff is the effective correlation length, i.e., a typical distance between the seeds of the new phase; and and d = 3, 2, and 1 for the monopoles, cosmic strings, and domain walls, respectively. (These concentrations refer evidently to unit volume, area, and length.) At last, substituting (2) into (1), we get: where H pt is the value of Hubble parameter at the instant of phase transition.
It should be mentioned that, unfortunately, the lower theoretical bound (3) is inconsistent with the upper bound following from observations (e.g., review [8]). One possible way to mitigate this disagreement is to modify a Lagrangian of the field theory under consideration, typically, by introduction of the "biased" vacuum (i.e., a priori removing the degeneracy) [4,5]. Yet another conceivable approach, which was not exploited before, is to take into account the nonlocal quantum correlations, which might manifest themselves in the macroscopic BEC.
However, exploiting the idea of macroscopic quantum correlations, one should bear in mind the following two subtle points: • Firstly, the most of laboratory studies of EPR correlations, performed since the 1970's, dealt with the microscopic quantum objects (e.g., atoms and photons). There were only a very few number of experiments on the nonlocal correlations in macroscopic systems; and they will be discussed in detail in the next section.
• Secondly, the correlations studied in microscopic objects were always associated with the exact conservation laws (most typically, the conservation of the total spin). In contract to these cases, the macroscopic BECs do not usually obey the suitable conservation laws. Nevertheless, it might be expected that correlations in the macroscopic systems could be caused just by the energetic criteria: if a correlated state of a large system possesses a less energy than its uncorrelated state, it should emerge with a greater probability. As will be shown in the next section, a number of recent laboratory experiments support this idea.

Review of the Laboratory Experiments
As far as we know, there are by now two groups of experiments confirming the phenomenon of nonlocal correlations in macroscopic BECs. The first of them are the experiments with superconducting multi-Josephson-junction loops (MJJL), which were started in the very beginning of 2000's [9]; and the second are the experiments with ultracold gases in the periodic optical potentials, which began a few years later [10,11].

MJJL Experiment
A general scheme of the original MJJL experiment [9] is shown in Figure 3: In the phase of cooling, when temperature T drops to T c = 90 K, the segments separated by the junctions become superconducting. However, the junctions themselves remain normal and, therefore, the superconducting segments are effectively separated from each other.
Therefore, a random phase of the superconducting order parameter should be established in each of them. (They are schematically illustrated in Figure 3 by the randomly oriented arrows.) The total phase variation of the order parameter along the loop, in general, should be nonzero.
At subsequent cooling down to the temperature T cJ , which is 5÷7 K below T c , the Josephson junctions become also superconducting. Consequently, due to the above-mentioned phase variation, a superconducting current I will develop along the loop. As a result, the loop will be penetrated by the magnetic flux Φ, which is just the measurable quantity. It should be mentioned that this setup is quite close to the original idea by Zurek [7], who proposed to observe a spontaneous rotation produced by a rapid phase transition to the In summary, because of the phase jumps between the isolated segments formed at the stage when T cJ < T < T c , the final magnetic flux Φ through the loop turns out to be nonzero and varies randomly from one cooling cycle to another. Its histogram N Φ (Φ) derived from a large number of cycles is well described by the normal (Gaussian) law with zero average value and standard deviation 7.4 φ 0 (where φ 0 is the magnetic flux quantum, and N Φ is the number of cases with the total magnetic flux Φ). This standard deviation is just the typical value of the flux spontaneously generated in one cooling cycle.
In fact, the above-written value is unreasonably large: if the phase jumps between the segments were absolutely independent of each other, then the expected width of the distribution would be only 3.6 φ 0 . On the other hand, this excessive value was satisfactorily explained by the authors of the experiment under assumption that phases of the superconducting order parameter in the isolated (i.e., "causally-disconnected") segments were correlated to each other, so that a probability P (δ i ) of the phase jump δ i in the i'th junction was given by the Gibbs law: where E J is the energy concentrated in the Josephson junction, T is the temperature, and k B is Boltzmann constant.
So, the main conclusion following from the above experiment is that the energy concentrated in the defects should be taken into account in the calculation of the probability of realization of various field configurations, even if the phase transformation develops independently in the remote parts of the system. For example, an array of 30 BECs of the ultracold gas in a regular one-dimensional lattice was created in the experiment [10]. Moreover, it was demonstrated that such condensates can well interfere with each other even if they were produced independently, i.e., "have never seen one another".

Experiments with Ultracold Gases in Optical Lattices
A next experiment of the same group [11] was devoted to a detailed study of the phase defects. In particular, an efficiency of the defect formation as function of the temperature was studied, and it was found that the number of the defects (dislocations) formed increases with temperature by qualitatively the same law as (4); see Figure 4 from the above-cited paper (a sharp outlier at c 0 = 0.2 is most probably an experimental error). 2 Therefore, both the MJJL and ultracold-atom experiments suggest that the probability of realization of various configurations (of the BEC order parameter) should be calculated taking into account the total energy of the system even if separate parts of this system do not interact with each other during a particular physical process (e.g., the phase transformation).
Of course, these parts of the system must be causally connected during its previous evolution.
(This is always satisfied in the laboratory experiments but requires a special consideration in the cosmological context.) In some sense, the above phenomenon may be considered as analog of EPR correlation for the system that does not posses an exact conservation law. In such a case, just the energetic criteria should come into play (see also discussion in the end of Section 1).

Cosmological Implications
There is evidently a close similarity between the symmetry-breaking phase transitions in the multi-Josephson-junction loop, depicted in Figure 3, and in the simplest one-dimensional (1D) Friedmann-Robertson-Walker (FRW) cosmological model, schematically illustrated in To make the quantitative estimates, let us consider the space-time metric where t is the time, x is the spatial coordinate, and a(t) is the scale factor of FRW model. This space-time is filled with the real scalar field ϕ, simulating the Higgs field in the theory of elementary particles. Its Lagrangian 2 In the analysis of Figure 4 from [11], it should be kept in mind that the parameter of interference contract c 0 , used as an independent variable in Plot 4c, is, roughly speaking, inversely proportional to the temperature. possesses Z 2 symmetry group, which should be broken by the phase transition.
As is known, the stable low-temperature vacuum states of the field (6) are and a transition region between them (domain wall) is described as Such a domain wall contains the energy From here on, we shall assume that thickness of the wall, ∼1/µ, is small in comparison with a characteristic distance between them; i.e., the domain walls will be treated as a point-like object.
Next, it is convenient to introduce the conformal time η = dt/a(t) . As a result, the space-time metric (5) will take the conformally flat form [12]: so that the light rays (ds 2 = 0) will represent the straight lines inclined at ±π/4 : The entire structure of the space-time can be conveniently described by the conformal diagram in Figure 5. Let η = 0 and η = η 0 be the beginning and end of the phase transition, respectively, and η = η * be the instant of observation. 3 Since it is commonly assumed that bubbles of the new phase grow at the rate close to the speed of light, their boundaries can be well depicted by the light rays. Then, as follows from a simple geometric consideration, is the number of spatial subregions in the observable Universe causally-disconnected during the phase transition. Their final vacuum states can be conveniently denoted by arrows, like spins.
Let us calculate a probability of the phase transition without formation of the domain walls in the observable space-time (the past light cone) P 0 N , where subscript N implies the number of subregions; and superscript 0, the absence of domain walls. A trivial estimate can be obtained by taking a ratio of the number of field configurations without domain walls (which equals 2) to their total number (2 N ): This quantity evidently tends to zero very quickly at large N. In other words, the observable part of the Universe, represented by the large upper triangle in Figure 5, will inevitably contain some number of the domain walls. Unfortunately, as was recognized long time ago [4,5], a presence of the domain walls is incompatible with astronomical observations.
A possible resolution of this paradox can be based just on taking into consideration the nonlocal Gibbs-like correlations (4). First of all, we must ensure that such correlations can really develop, i.e., the subdomains of new phase were causally connected in the past. As is seen in Figure 5, this is really possible if a sufficiently long interval of the conformal time preceded the phase transition. Then, the lower shaded triangle will cover at the instant η = 0 the upper triangle, representing the observable part of he Universe.
The inequality (14) can be satisfied, particularly, in the case of sufficiently long inflation- so that the above-mentioned interval ∆η can be sufficiently large. Let us remind that, from the viewpoint of elementary-particle physics, the de Sitter stage can be naturally realized in the overcooled state of the Higgs field immediately before its first-order phase transition; and just this idea was the starting point of the first inflationary models [13].
Next, if the condition (14) is satisfied, then it is reasonable to assume that the abovementioned correlations (4) should take place between the all N subdomains drawn in the conformal diagram ( Figure 5). 4 In such a case, the probability P 0 N should be calculated taking into account Gibbs factors for the field configurations involving domain walls: where Here, s j is the spin-like variable denoting a sign of the vacuum state in the j'th subdomain, E is the domain wall energy, given by (9), and T is the characteristic temperature of the phase transition. (From here on, the temperature will be expressed in energetic units; so the Boltzmann constant k B will be omitted.) From a formal point of view, statistical sum (17) is very similar to the sum for Ising model, well studied in the physics of condensed matter, e.g. [15]. Using exactly the same mathematical approach, 5 we get the final result: (Yet another method for calculation of the same quantity, which is more straightforward and pictorial but less informative, can be found in [14]; the approach outlined here was employed for the first time in our paper [16].) The quantity P 0 N as function of N and E/T is plotted in Figure 6. It is seen that P 0 parameter E/T is sufficiently large. Some other plots illustrating suppression of concentration of the domain walls by the nonlocal correlations can be found in paper [17], devoted to the phase transformations in superfluids and superconductors. Therefore, just the large energy concentrated in the domain walls turns out to be the factor substantially reducing the probability of their creation.
Next, as can be easily derived from (18), the probability of absence of the domain walls in the observable Universe becomes on the order of unity, e.g. 1/2, if E/T ln N. Taking into account (9) and (12), this inequality can be rewritten as Because of the very weak logarithmic dependence in the right-hand side, such a condition could be reasonably satisfied for a certain class of field theories.
Moreover, the situation becomes even more favorable in the case of two or three spatial dimensional space. The point is that a well-known property of the 2D and 3D Ising models is a tendency for aggregation of the domains with the same value of the order parameter when the temperature drops below some critical value T c ∼ E [18]. In the condensed-matter applications, this corresponds, for example, to the spontaneous magnetization of a solid body. Regarding the cosmological context, one can expect that probability of formation of the domain walls will be reduced dramatically at the sufficiently large values of E/T ; some illustrations of this phenomenon can be found in [17]. (To avoid misunderstanding, let us emphasize that the above-mentioned Ising models are only the auxiliary mathematical constructions, describing a final distribution of the domain walls after the phase transformation. So, the formal phase transitions in the 2D and 3D Ising models should not be associated with the physical phase transition in the original ϕ 4 -field model (6); for more details, see Table 2 in [17].)

Discussion and Conclusions
As was discussed in the present article, a few laboratory experiments suggest a presence of the nonlocal Gibbs-like correlations between the phases of BECs after the rapid phase transformations. Therefore, it can be reasonably conjectured that the same correlations should occur in the BEC of Higgs field, which is formed in the course of evolution of the Universe. As follows from our quantitative estimates for the simplest case of 1D FRW cosmological model, such correlations can show a way to resolve the well-known problem of the excessive concentration of the domain walls resulting from the phase transitions in the early Universe. In fact, this problem was recognized in the mid 1970s [4]; and since that time a commonly-used approach to resolve the domain wall problem was based on the introduction of the so-called "biased" (or asymmetric) vacuum. As a result, under the appropriate choice of parameters, the regions of "false" (energetically unfavorable) vacuum should quickly disappear, eliminating the corresponding domain walls [5]. Unfortunately, the concept of biased vacuum was not supported by independent data in the physics of elementary particles. On the other hand, the idea of nonlocal correlations, employed in the present study, is supported by at least a few laboratory experiments. Therefore, it looks more attractive from our point of view.
Yet another cosmological problem, recognized in the recent decade, is the anomalous behaviour of fluctuations in the cosmic microwave background radiation (CMBR) at large angular scales, approximately over 10 o [19,20]. It can be conjectured that such anomalies are also associated with the effect of the nonlocal correlations in the early Universe; but, of course, a much more elaborated analysis must be performed to to draw any reasonable conclusions.
At last, we would like to mention a recent activity in the experiments with ultracold cases for simulation of various dynamical phenomena in cosmology, e.g., the so-called Sakharov oscillations [21]. From our point of view, a more careful study of the nonlocal correlations may become an important branch in this rapidly-growing research field.