aa r X i v : . [ g r- q c ] A p r Nonlocal Quantum Effects in Cosmology
Yurii V. Dumin
1, 2, ∗ Sternberg Astronomical Institute (GAISh) ofLomonosov Moscow State University,Universitetski prosp. 13, 119991, Moscow, Russia Space Research Institute (IKI) ofRussian Academy of Sciences,Profsoyuznaya str. 84/32, 117997, Moscow, Russia (Dated: January 7, 2014; Revised: April 7, 2014)
Abstract
Since it is commonly believed that the observed large-scale structure of the Universe is an imprintof quantum fluctuations existing at the very early stage of its evolution, it is reasonable to pose thequestion: Do the effects of quantum nonlocality, which are well established now by the laboratorystudies, manifest themselves also in the early Universe? We try to answer this question by utilizingthe results of a few experiments, namely, with the superconducting multi-Josephson-junction loopsand the ultracold gases in periodic potentials. Employing a close analogy between the above-mentioned setups and the simplest one-dimensional Friedmann–Robertson–Walker cosmologicalmodel, we show that the specific nonlocal correlations revealed in the laboratory studies mightbe of considerable importance also in treating the strongly-nonequilibrium phase transitions ofHiggs fields in the early Universe. Particularly, they should substantially reduce the number oftopological defects ( e.g. , domain walls) expected due to independent establishment of the newphases in the remote spatial regions. This gives us a hint for resolving a long-standing problemof 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 cosmicmicrowave background fluctuations at large angular scales. ∗ E-mail: [email protected], [email protected] . 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 mea-surements of two physical objects located in the causally-disconnected regions of space, i.e. ,beyond the light cones of each other. In the modern and most frequently used in the exper-iments form, this phenomenon can be illustrated in Figure 1. Here, an original particle ofzero spin decays at the instant t = 0 into two particles with equal but oppositely-directedspins s and s , which subsequently move from each other in the opposite directions. Next,if measurements of the spins of both particles are performed in the remote spatial points x and x at the same instant of time, their values turn out to be perfectly correlated ( s = − s )just because of the law of spin conservation.At first sight, such correlation is quite surprising, because the measurements are per-formed in the spots of space–time lying beyond the mutual light cones ( i.e. , in the causallydisconnected regions). However, the existence of EPR correlations is well confirmed now bya lot of laboratory experiments. In fact, these correlations look much less surprising if wekeep in mind the fact that both light cones include the same common source in the past.It might be reasonable to emphasize also that, despite of a “superluminal” character of theEPR correlations, they cannot be employed for a faster-than-light communication, becausethe outcomes of correlated measurements of s and s in the points x and x are random.Turning attention to cosmology, we should first of all mention that it is widely believed xx s s x t Source pastlight conefuture lightcone future lightconepastlight cone
Detectors
EPR correlation
FIG. 1: Sketch of the typical laboratory ERP experiment. i.e. , a macroscopic quantumstate, in which the specific quantum correlations may naturally occur.The most important feature in temporal dynamics of the Higgs field is phase transitioncaused by the evolving temperature of the Universe, which can finally result in the formationof the nontrivial states of the physical vacuum [2]. In fact, the problem of complex vacuumwas recognized long time ago, just after appearance of the idea of spontaneous symmetrybreaking in the quantum field theory. Particularly, at the Conference on the occasion of the400th anniversary of Galileo Galilei’s birth, held in Pisa in 1964, Bogoliubov emphasizedthat “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]. Inthe next decade, the problem of formation of the domain walls in the course of cosmologicalevolution 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 domainwall dynamics was given, for example, in paper [5].)A commonly-accepted scenario of formation of the domain walls by the phase transitionin the early Universe is illustrated in Figure 2. A uniform initial (“symmetric”) state of theHiggs 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 ofa 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 originateat the same instant of time t = 0.) Since the seeds of the low-temperature phase emergeindependently in the remote spatial regions, 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 theinstant t = t collide with each other. However, if the values of the symmetry-broken phasein two neighboring domains were different, they cannot merge smoothly. Instead, a stabletopological defect—domain wall (or “kink”)—should be formed at their boundary. So, if the3 nstant ofobservationend ofphase transitiononset ofphase transitionBig Bangsymmetric stateof the fieldsymmetry-brokenstate of the fielddomainsof variousphases domainwalls seedsof new phaseobservablepart of theUniverse r ξ tt t * FIG. 2: Sketch of development of the phase transition in the expanding Universe. initial separation between the seeds of the new phase was ξ , then the resulting concentrationof the domain walls can be roughly estimated as n ≈ /ξ . This is the so-called Kibble–Zurek(KZ) mechanism for the formation of topological defects after the strongly-nonequilibriumphase transformations [6, 7].Strictly speaking, the scenario outlined in Figure 2 refers to the simplest case of Higgsfield, possessing Z symmetry group ( i.e. , admitting a discrete symmetry breaking). How-ever, the same basic idea is applicable also to more realistic Higgs fields with the continuoussymmetries, whose breaking can lead to the formation of more complex defects of the vac-uum, such as the monopoles and cosmic strings (or vortices). In general, KZ mechanismgives the following estimate for concentration of the topological defects: n ≈ / ξ d eff , (1)where ξ eff is the effective correlation length, i.e. , a typical distance between the seeds ofthe new phase; and d = 3, 2, and 1 for the monopoles, cosmic strings, and domain walls,respectively. (Their concentrations refer evidently to unit volume, area, and length.)An accurate calculation of the effective correlation length, in general, is a difficult task.However, its upper bound can be obtained from a simple causality argument: ξ eff ≤ c t pt ,where c is the speed of light, and t pt is the characteristic time from the Big Bang to the4nstant of phase transition. Next, the time from the Big Bang in Friedmann–Robertson–Walker (FRW) cosmology is estimated as ∼ /H , where H is the value of Hubble parameterat the respective instant. Consequently, ξ eff . c /H pt . (2)At last, substituting (2) into (1), we get: n & ( H pt /c ) d , (3)where H pt is the value of Hubble parameter at the instant of phase transition. Unfortunately, the lower theoretical bound (3) is inconsistent with the upper boundsfollowing from observations ( e.g. , review [8]). One possible way to mitigate this disagreementis to modify Lagrangian of the field theory under consideration, typically, by introductionof the “biased” vacuum (thereby, a priori removing the degeneracy) [4, 5]. Yet anotherconceivable approach, which was not exploited before, is to take into account the nonlocalquantum correlations, which might manifest themselves in the macroscopic BEC.However, exploiting the idea of macroscopic quantum correlations, one should bear inmind 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 wereonly a very few number of experiments on the nonlocal correlations in macroscopicsystems; and they will be discussed in detail in the next section. • Secondly, the correlations studied in microscopic objects were always associated withthe exact conservation laws (most typically, the conservation of the total spin). Incontract to these cases, the macroscopic BECs do not usually obey the suitable con-servation laws. Nevertheless, it might be expected that correlations in the macroscopicsystems could be caused just by the energetic criteria: if correlated state of a largesystem possesses less energy than its uncorrelated state, then it should emerge with a To avoid misunderstanding, let us mention that most of laboratory experiments aimed at verificationof KZ mechanism measured a scaling relationship between the size of uniform domains and the quenchrate, rather than the absolute number of the defects. However, in cosmological applications it is moreappropriate to discuss just the absolute concentration of the defects.
2. Review of the Laboratory Experiments
As far as we know, there are by now two groups of experiments confirming the phe-nomenon of nonlocal correlations in macroscopic BECs. The first of them are the experi-ments with superconducting multi-Josephson-junction loops (MJJL), which were started inthe very beginning of 2000’s [9]; and the second are the experiments with ultracold gases inthe periodic optical potentials, which began a few years later [10, 11].
A general scheme of the original MJJL experiment [9] is shown in Figure 3: A thin quasi-one-dimensional loop was fabricated from the YBa Cu O δ superconductor and contained214 segments separated from each other by the Josephson junctions (which are the micro-scopic domains of the same substance but with a higher critical temperature due to defectsin the crystalline lattice). The experimental procedure consisted of the numerous cycles ofvery quick cooling (approximately from 100 K to 77 K) and subsequent heating of the loopand measurement of the resulting electromagnetic response.In the phase of cooling, when temperature T drops approximately to T c = 90 K, the seg-ments separated by the junctions become superconducting. However, the junctions them-selves remain normal and, therefore, the superconducting segments are effectively separatedfrom each other. Consequently, a random phase of the superconducting order parametershould be established in each of them. (They are schematically illustrated in Figure 3 bythe randomly oriented arrows.) The total phase variation of the order parameter along theloop, in general, should be nonzero.At subsequent cooling down to the temperature T cJ , which is 5 ÷ T c , the Joseph-son junctions become also superconducting. Consequently, due to the above-mentionedphase variation, a superconducting current I will develop along the loop. As a result, the The actual loop used in the experiment was not perfectly circular, as in figure, but represented a windingstrip engraved in the superconductor film. T > c T T < cJ T T T cJ < < c Random phases
Correlated phases
Φ Φ N Φ φ N Φ φ ÷ I Φ FIG. 3: Basic design and principal results of the MJJL experiment. loop will be penetrated by the magnetic flux Φ, which is just the measurable quantity. Itshould be mentioned that this setup is quite close to the original idea by Zurek [7], whoproposed to observe a spontaneous rotation produced by a rapid phase transition to thesuperfluid phase in a thin annular tube. Unfortunately, such an experiment was never im-plemented in practice, since it is hardly possible to observe a mechanical rotation of themacroscopic body with angular momentum of just a few quanta ~ . On the other hand, themagnetic flux measurements by modern apparatus can easily detect the individual quantaof the magnetic flux. Therefore, an electromagnetic analog of the original Zurek’s proposalbecame feasible.In summary, because of the phase jumps between the isolated segments formed at thestage when T cJ < T < T c , the final magnetic flux Φ through the loop turns out to benonzero and varies randomly from one heating–cooling cycle to another. The 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 . φ (where φ is the magnetic fluxquantum, and N Φ is the number of cases with the total magnetic flux Φ). This standard7eviation is just the typical value of the flux spontaneously generated in one cycle.In fact, the above-written value is unreasonably large: if the phase jumps between thesegments were absolutely independent of each other, then the expected width of the distri-bution would be only 3 . φ . However, the excessive value was satisfactorily explained bythe authors of the experiment under assumption that phases of the superconducting orderparameter in the isolated ( i.e. , “causally-disconnected”) segments were correlated to eachother, so that probability P ( δ i ) of the phase jump δ i in the i ’th junction was given by theGibbs law: P ( δ i ) ∝ exp[ − E J ( δ i ) /k B T ] , (4)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 concen-trated in the defects should be taken into account in the calculation of the probability ofrealization of various field configurations, even if the phase transformation develops inde-pendently in the remote parts of the system. The MJJL experiment, discussed in the previous section, gave the first hint to the im-portance of using the Gibbs law even for the systems composed of the apparently isolatedparts. Unfortunately, this experiment did not enable us to check the particular functionaldependence (4). (To do so, it would be necessary to perform the same experiment withdifferent types of superconductors, which has not been fulfilled by now.)Nevertheless, a few yeas later it became possible to solve this tack by using the BECsof ultracold gases in periodic potentials (or the so-called optical lattices), formed by theintersecting laser beams. These systems represent a close analog of the multiple Josephsonjunctions and, as distinct from the solid-state setups, their parameters can be easily varied.A diagnostics of the phase jumps in such installations is performed by a removal of theexternal potential, thereby enabling the pieces of BEC to expand and interfere with eachother.For example, an array of 30 BECs of the ultracold gas in a regular one-dimensional latticewas created in the experiment [10]. Next, it was demonstrated that such condensates can8 .1Average contract of interference pattern, c F r a c t i on o f i m age s w i t hd i s l o c a t i on s T FIG. 4: Fraction of the interference patterns showing at least one dislocation as function of theinterference contrast c . Inset, examples of images with the increasing number of dislocations(from bottom to top). Adapted by permission from Macmillan Publishers Ltd: Nature , vol. 441,no. 7097, pp. 1118–1121, c (cid:13) well interfere with each other even if they were produced independently, i.e. , “have neverseen one another”.A further experiment of the same group [11] was devoted to a detailed study of thephase defects. In particular, an efficiency of the defect formation was measured as functionof temperature. (In fact, the determination of temperature is not an easy task in theexperiments with ultracold gases. So, the primary independent parameter was taken to bethe average contract of the interference pattern c , which is, roughly speaking, inverselyproportional to the temperature T .) As a result, it was found that the number of defects(dislocations) formed in the BEC of ultracold gas increases with temperature by qualitativelythe same law as (4); see Figure 4 (a sharp outlier at c ≈ . the total energy of the system even if separate parts of this system do not interactwith each other during a particular physical process ( e.g. , the phase transformation). Of9 a t ( ) Symmetricphase Symmetry-brokenphase
FIG. 5: Sketch of the phase transition in 1D FRW cosmological model (the physical distance ismeasured along the circles). 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 inthe cosmological context.In some sense, the above phenomenon can be interpreted as analog of EPR correlation forthe system that does not posses an exact conservation law. In such a case, just the energeticcriteria should come into play (see also discussion in the end of Section 1).
3. Cosmological Implications
There is evidently a close similarity between the symmetry-breaking phase transitions inthe multi-Josephson-junction loop, depicted in Figure 3, and in the simplest one-dimensional(1D) Friedmann–Robertson–Walker (FRW) cosmological model, schematically illustrated inFigure 5. For the sake of definiteness, we shall consider only the simplest type of defects,namely, the domain walls or kinks.To make the quantitative estimates, let us consider the space–time metric ds = dt − a ( t ) dx , (5)where t is the time, x is the spatial coordinate, and a ( t ) is the scale factor of FRW model.10From here on, we shall assume that c ≡ ϕ , simulating Higgs field in the theory of elementary particles. Its Lagrangian L ( x, t ) = 12 (cid:2) ( ∂ t ϕ ) − ( ∂ x ϕ ) (cid:3) − λ (cid:2) ϕ − (cid:0) µ /λ (cid:1) (cid:3) (6)possesses Z symmetry group, which should be broken by the phase transition.As is known, the stable low-temperature vacuum states of the field (6) are ϕ = ± µ / √ λ , (7)and a transition region between them (domain wall) is described as ϕ ( x ) = ϕ tanh (cid:2) ( µ/ √ x − x ) (cid:3) . (8)Such a domain wall contains the energy E = 2 √ µ λ . (9)We shall assume that thickness of the wall, ∼ /µ , is small in comparison with a characteristicdistance between them; i.e. , the domain walls can be treated as point-like objects.Next, it is convenient to introduce the conformal time η = R dt/a ( t ) . As a result, thespace–time metric (5) will take the conformally flat form [12]: ds = a ( t ) [ dη − dx ] ; (10)so that the light rays ( ds = 0) will represent the straight lines inclined at ± π/ x = ± η + const . (11)The entire structure of the space–time can be conveniently described by the conformaldiagram in Figure 6. Let η = 0 and η = η be the beginning and end of the phase transition,respectively, and η = η ∗ be the instant of observation. Since it is commonly assumed thatbubbles of the new phase grow at the rate close to the speed of light, their boundaries canbe well depicted by the light rays. Then, as follows from a simple geometric consideration, N = ( η ∗ − η ) /η ≈ η ∗ /η (at large N ) (12) The instants η = 0, η , and η ∗ of the conformal time correspond to the instants t = 0, t , and t ∗ of thephysical time in Figure 2. x η ∆ ηη η ∗ FIG. 6: Conformal diagram of 1D FRW cosmological model. 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 the arrows,like spins.Let us calculate a probability of the phase transition without formation of the domainwalls in the observable space–time (the past light cone) P N , where subscript N implies thenumber of subregions; and superscript 0, the absence of domain walls. A trivial estimatecan be obtained by taking a ratio of the number of field configurations without domainwalls (which equals 2) to their total number (2 N ): P N = 2 / N = 1 / N − . (13)This quantity evidently tends to zero very quickly at large N . In other words, the observablepart of the Universe, represented by the large upper triangle in Figure 6, will inevitablycontain some number of the domain walls. Unfortunately, as was recognized long timeago [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 thenonlocal Gibbs-like correlations (4). First of all, we must ensure that such correlations canreally develop, i.e. , the subdomains of the new phase were causally connected in the past.As is seen in Figure 6, this is really possible if a sufficiently long interval of the conformal12ime ∆ η ≥ η ∗ (14)preceded the phase transition. Then, the lower shaded triangle will cover at the instant η = 0the upper triangle, representing the observable part of the Universe.The inequality (14) can be satisfied, particularly, in the case of sufficiently long inflation-ary (de Sitter) stage. Really, if a ( t ) ∝ exp( Ht ), where H is Hubble constant, then η ∝ − H e − Ht + const → −∞ at t → −∞ ; (15)so that the above-mentioned interval ∆ η can be sufficiently large. Let us remind that, fromthe viewpoint of elementary-particle physics, the de Sitter stage can be naturally realizedin 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 above-mentioned correlations (4) should take place between the all N subdomains drawn in theconformal diagram, Figure 6. (It is interesting to mention that in our old work [14], per-formed before the MJJL experiment, the same Gibbs-like correlations were introduced onthe basis of some metaphysical speculations.) In such a case, the probability P N shouldbe calculated taking into account Gibbs factors for the field configurations involving thedomain walls: P N = 2 / Z , (16)where Z = N X i =1 X s i = ± exp (cid:26) − ET N X j =1
12 (1 − s j s j +1 ) (cid:27) . (17)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 thephase transition. (From here on, the temperature will be expressed in energetic units; sothe Boltzmann constant k B will be omitted.)From a formal point of view, statistical sum (17) is very similar to the sum for Isingmodel, well studied in the physics of condensed matter, e.g. [15]. Using exactly the same13 N E T / P N FIG. 7: The probability of phase transition without formation of the domain walls P N as functionof the number of disconnected subregions N and the ratio of the domain wall energy to the phasetransition temperature E/T . mathematical approach, we get the final result: P N = 2[1 + e − E/T ] N + [1 − e − E/T ] N . (18)(Yet another method for calculation of the same quantity, which is more straightforward andpictorial but less informative, can be found in [14]; the approach outlined here was employedfor the first time in our paper [16].)The quantity P N as function of N and E/T is plotted in Figure 7. It is seen that P N drops very sharply with increase in N at small values of E/T (when the effect of Gibbs-likecorrelations is insignificant), but it becomes a gently decreasing function of N when theparameter E/T is sufficiently large. Some other plots illustrating suppression of concentra-tion of the domain walls by the nonlocal correlations can be found in paper [17], devotedto the phase transformations in superfluids and superconductors. Therefore, just the largeenergy concentrated in the domain walls turns out to be the factor substantially reducingthe probability of their creation.Next, as can be easily derived from (18), the probability of absence of the domain walls Attention should be paid to the appropriate choice of zero energy, which is different from the one commonlyused in the condensed-matter physics.
14n the observable Universe becomes on the order of unity, e.g.
E/T & ln N . Takinginto account (9) and (12), this inequality can be rewritten as µ λ T & ln η ∗ η . (19)Because of the very weak logarithmic dependence in the right-hand side, such a conditioncould be reasonably satisfied for a certain class of field theories.Moreover, the situation becomes even more favorable in the case of two- or three-dimensional space. The point is that a well-known property of the 2D and 3D Ising modelsis a tendency for aggregation of the domains with the same value of the order parameterwhen the temperature drops below some critical value T c ∼ E [18]. In the condensed-matterapplications, this corresponds, for example, to the spontaneous magnetization of a solidbody. Regarding the cosmological context, one can expect that probability of formation ofthe domain walls will be reduced dramatically at the sufficiently large values of E/T ; someillustrations of this phenomenon can be found in [17]. (To avoid misunderstanding, let usemphasize that the above-mentioned Ising models are only the auxiliary mathematical con-structions, 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 associatedwith the physical phase transition in the original ϕ -field model (6); for more details, seeTable 2 in [17].)
4. Discussion and Conclusions
As was discussed in the present article, a few laboratory experiments suggest a presenceof the nonlocal Gibbs-like correlations between the phases of BECs after the rapid phasetransformations. Therefore, it can be reasonably conjectured that the same correlationsshould occur in the BEC of Higgs field, which is formed in the course of evolution of theUniverse. As follows from our quantitative estimates for the simplest case of 1D FRW cos-mological model, such correlations can show a way to resolve the well-known problem ofthe excessive concentration of the domain walls resulting from phase transitions in the earlyUniverse. In fact, this problem was recognized in the mid 1970s [4]; and since that time acommonly-used approach to its resolution was based on the introduction of the so-called “bi-ased” (or asymmetric) vacuum. As a result, under the appropriate choice of parameters, the15egions of “false” (energetically unfavorable) vacuum should quickly disappear, eliminatingthe corresponding domain walls [5]. Unfortunately, the concept of biased vacuum was notsupported 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 afew laboratory experiments. Therefore, from our point of view, it looks more attractive.It should be emphasized again that a number of arguments from the modern observationalcosmology impose severe constraints on the concentration of domain walls. In particular, anappreciable number of the domain walls would produce an unacceptable anisotropy of thecosmic microwave background (CMB) radiation, change the overall rate of the cosmologicalexpansion, etc . Besides, it is commonly believed now that the primordial spectrum ofdensity fluctuations, responsible for the formation of the large-scale structure, is formed bythe Gaussian quantum fluctuations in the very early Universe amplified by the subsequentinflationary stage, while contribution from the topological defects is quite insignificant. Allthese facts imply that there should be an efficient mechanism for the suppression of thedomain walls, and the nonlocal quantum correlations discussed in the present paper mightbe a reasonable option.Yet another recent cosmological problem, recognized due to
WMAP and confirmed bythe
Planck satellite data, is the anomalous behaviour of CMB fluctuations at large angularscales, approximately over 10 o [19, 20]. It can be conjectured that such anomalies areassociated just with the nonlocal correlations in the early Universe; but, of course, a muchmore elaborated analysis must be performed to draw a reliable conclusion.At last, we would like to mention a recent activity in the experiments with ultracold gasesfor simulation of various dynamical phenomena in cosmology, e.g. , the so-called Sakharovoscillations [21]. From our point of view, a more careful study of the nonlocal correlationsmay become an important branch in this rapidly-growing research field. Acknowledgments
The initial stage of this work was substantially supported by the ESF COSLAB (Cos-mology in the Laboratory) Programme.I am grateful to a large number of researchers with whom I discussed the problem ofnonlocal effects in cosmology since the early 2000s till now: E. Arimondo, V.B. Belyaev,16.A. Bertlmann, Yu.M. Bunkov, A.M. Chechelnitsky, I. Coleman, J. Dalibard, V.B. Efimov,V.B. Eltsov, H.J. Junes, I.B. Khriplovich, T.W.B. Kibble, M. Knyazev, V.P. Koshelets,O.D. Lavrentovich, V.N. Lukash, A. Maniv, P.V.E. McClintock, L.B. Okun, G.R. Pick-ett, E. Polturak, A.I. Rez, R.J. Rivers, M. Sakellariadou, M. Sasaki, R. Sch¨utzhold,V.B. Semikoz, M. Shaposhnikov, A.A. Starobinsky, A.V. Toporensky, W.G. Unruh, G. Vi-tiello, G.E. Volovik, C. Wetterich, and W.H. Zurek.The author declares that there is no conflict of interests regarding the publication of thisarticle. [1] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physicalreality be considered complete?”
Physical Review , vol. 47, no. 10, pp. 777–780, 1935.[2] A.D. Linde, “Phase transitions in gauge theories and cosmology,”
Reports on Progress inPhysics , vol. 42, no. 3, pp. 389–437, 1979.[3] N.N. Bogoliubov, “Field-theoretical methods in physics,”
Supplemento al Nuovo Cimento(Serie prima) , vol. 4, no. 2, pp. 346–357, 1966.[4] Ia.B. Zeldovich, I.Yu. Kobzarev, and L.B. Okun, “Cosmological consequences of a spontaneousbreakdown of a discrete symmetry,”
Soviet Physics—JETP , vol. 40, no. 1, pp. 1–5, 1975[Translated from:
Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki , vol. 67, p. 3–11, 1974].[5] G.B. Gelmini, M. Gleiser, and E.W. Kolb, “Cosmology of biased discrete symmetry breaking,”
Physical Review D , vol. 39, no. 6, pp. 1558–1566, 1989.[6] T.W.B. Kibble, “Topology of cosmic domains and strings,”
Journal of Physics A: Mathemat-ical and General , vol. 9, no. 8, pp. 1387–1398, 1976.[7] W.H. Zurek, “Cosmological experiments in superfluid helium?”
Nature , vol. 317, no. 6037,pp. 505–508, 1985.[8] H.V. Klapdor-Kleingrothaus and K. Zuber,
Particle Astrophysics , Institute of Physics Pub-lishing, Bristol, 1997.[9] R. Carmi, E. Polturak, and G. Koren, “Observation of spontaneous flux generation in a multi-Josephson-junction loop,”
Physical Review Letters , vol. 84, no. 21, pp. 4966–4969, 2000.[10] Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and J. Dalibard, “Interference of an arrayof independent Bose–Einstein condensates,”
Physical Review Letters , vol. 93, no. 18, Article D 180403 (4 pp.), 2004.[11] Z. Hadzibabic, P. Kruger, M. Cheneau, B. Battelier, and J. Dalibard, “Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas,”
Nature , vol. 441, no. 7097, pp. 1118–1121, 2006.[12] C.W. Misner, “Mixmaster Universe,”
Physical Review Letters , vol. 22, no. 20, pp. 1071–1074,1969.[13] A.D. Linde, “The inflationary Universe,”
Reports on Progress in Physics , vol. 47, no. 8,pp. 925–986, 1984.[14] Yu.V. Dumin, “On a probable role of EPR (Einstein–Podolsky–Rosen) correlations in breakingthe symmetry of Higgs fields in cosmological phase transitions,”
Hot Points in Astrophysics:Proceedings of the International Workshop , pp. 114–120, Joint Institute for Nuclear Research,Dubna, 2000.[15] A. Isihara,
Statistical Physics , Academic Press, New York, 1971.[16] Yu.V. Dumin, “On the influence of Einstein–Podolsky–Rosen effect on the domain wall for-mation during the cosmological phase transitions,”
Frontiers of Particle Physics: Proceedingsof the Tenth Lomonosov Conference on Elementary Particle Physics , pp. 289–294, WorldScientific Publishing Co., Singapore, 2003.[17] Yu.V. Dumin, “Ultracold gases and multi-Josephson junctions as simulators of out-of-equilibrium phase transformations in superfluids and superconductors,”
New Journal ofPhysics , vol. 11, no. 10, Article ID 103032 (12 pp.), 2009.[18] Yu.B. Rumer and M.Sh. Ryvkin,
Thermodynamics, Statistical Physics, and Kinetics , Mir,Moscow, 1980.[19] A. Wright, “Across the Universe,”
Nature Physics , vol. 9, no. 5, p. 264, 2013.[20] R. Hofmann, “The fate of statistical isotropy,”
Nature Physics , vol. 9, no. 11, pp. 686–689,2013.[21] J. Schmiedmayer and J. Berges, “Cold atom cosmology,”
Science , vol. 341, no. 6151, pp. 1188–1189, 2013., vol. 341, no. 6151, pp. 1188–1189, 2013.