Possible formation of ring galaxies by torus-shaped magnetic wormholes

We present the hypothesis that ring galaxies were formed by relic magnetic torus - shaped wormholes. In the primordial plasma before the recombination magnetic fields of wormholes trap baryons whose energy is smaller than a threshold energy. They work as the Maxwell's demons collecting baryons from the nearest (horizon size) region and thus forming clumps of baryonic matter which have the same torus-like shapes as wormhole throats. Afterwards the clumps serve as seeds for the formation of ring galaxies. Upon the recombination torus-like clumps may decay and merge leading to spirals and ellipticals. We show that there are threshold values of magnetic fields which give the upper and lower boundary values for the baryon clumps in such protogalaxies.


I. INTRODUCTION
The widely accepted theory of galaxy formation is based on Lambda-CDM model. The primordial inhomogeneities in dark matter develop and first form relatively small clumps of matter which eventually merge and form galaxies and groups. This picture represents the hierarchical process [1,2] and the theory is heavily based on the presence of sufficiently strong inhomogeneities in cold dark matter (δρ DM /ρ DM ∼ 10 −3 at the moment of the recombination). It is rather successful and only slightly underestimates the number of thin disk galaxies in the universe. It is recognized that upon some refining it allows to reproduce the population of galaxies in our Universe.
However, some part of galaxies are shaped like a doughnut, the so-called ring galaxies [3,4], and may contain a hole in the middle, e.g., see the recently found giant ring galaxy (ID 5519) [5]. It is rather difficult to organize such structures in a natural way within the hierarchical process. Some of such galaxies have neither an accompanying galaxy to be produced by the scattering process, nor a collection of stars in the center of the ring to produce radial density waves. This allows us to present the hypothesis that all such ring galaxies are remnants of relic magnetic wormholes.
Wormholes are exotic objects whose existence is predicted by general relativity (GR), e.g. see the history in Refs. [6]. Enormous efforts have been made to construct exact solutions of the Einstein equations which correspond to a stable wormhole, e.g., see some examples in Refs [7]. The dry rest is that in the case of spherically symmetric configurations wormholes are highly unstable and collapse very rapidly. To be stable they require the presence of exotic (violating the weak energy condition) matter. This means that all primordial spherical wormholes have collapsed long ago and they cannot be distinguished from black holes [33].
Over the last decade there is an essential increase of * Electronic address: ka98@mail.ru † Electronic address: savelova@bmstu.ru interest in different modifications of GR in which the violating energy condition matter is replaced by an appropriate modification e.g., see Refs [8][9][10]. We leave aside such a possibility and explore less symmetric configurations. It turns out that less symmetric configurations can be made stable without exotic matter or any modification of GR. First rigorous example was presented in Ref. [11]. It was demonstrated that in the open Friedman model a stable wormhole can be obtained simply by the factorization of space over a discrete subgroup of the group of motions of space [34]. We point out that such a procedure does not allow the spherical configurations at all. The simplest wormhole obtained by the factorization has the throat in the form of a torus, i.e., it has the shape of a doughnut [12]. We stress that such wormholes are not static but expand in agreement with the expansion of the Universe. They are frozen into space and, therefore, are static in the co-moving coordinates. It was also demonstrated that the factorization allows to get an arbitrary number of such wormholes in space.
In flat space the torus -like wormholes become dynamical objects and evolve [11]. Whether they are static, expand, or collapse, depends on surrounding matter and peculiar motions of throats. The shape of throats of such wormholes resembles a doughnut and is characterized by two radii R w and r w . In the limit R w ≫ r w it can be approximately described by the cylindrical (axial) configuration. It turns out that static and stationary cylindrical wormhole solutions do exist and it was found that asymptotically flat wormhole configurations do not require exotic matter violating the weak energy condition [13,14]. Such encouraging results show that such objects, as doughnut -shaped wormholes, have all chances to be observed in astrophysical systems. We stress that the complete description of the evolution of such wormholes in flat space represents a rather complex problem which still awaits for the rigorous investigation.
The possibility to directly observe wormholes attracts the more increasing attention, e.g., see Refs. [15][16][17][18]. At first glance the most promising are collective effects produced by a distribution of wormholes in space. However, our previous investigation have shown that observational effects of a distribution of wormholes are very well hidden under analogous effects produced by ordinary matter e.g., see Refs. [19,20]. The only exclusion may be the noise (stochastic background) produced by the scattering of emitted by binaries gravitational waves on wormholes [21].
In general a single wormhole produces much less noticeable effects (lensing, cosmic ray scattering, etc.). However, wormholes may possess non-trivial magnetic fields as vacuum solutions. In this case possible imprints of wormholes in the present picture of the Universe may be rather considerable. In particular, when such a magnetic wormhole gets close to a galaxy, it starts to work as an accelerator of charged particles [12] which is capable of explaining the origin of high-energy cosmic-ray particles [22]. In voids, such a wormhole works simply as a generator of synchrotron radiation and can be detected via the magnetic field [23,24]. The primordial magnetic fields [25] in turn may form small-scale nonlinear clumps of baryonic matter [26,27] and as it was recently shown [28] they allow to solve the existing tension between the Hubble constant value measured by Planck H 0 = 67.36±0.54km s −1 M pc −1 [29] and measured by the Supernovae H 0 = 74.03±1.42kms −1 M pc −1 [30]. In other words, magnetic wormholes should leave a clear imprint on the sky. It turns out that relic magnetic wormholes may play also the key role in formation of ring galaxies.

II. MAGNETIC WORMHOLES AS BARYON TRAPS
Consider a single wormhole whose throat has the shape of a torus (the genus -1 wormhole by the classification suggested in Ref. [12]). In the presence of such a wormhole Maxwell's equations possess two additional classes of non-trivial vacuum solutions. Indeed, according to the Stocks theorem the system of vacuum Maxwell equations implies Bdl = 0 for any loop which can be pulled to a point (where B is the magnetic field). In the case of a non-trivial topology of space[35] there appear new classes of loops Γ a which cannot be contracted to a point and, therefore, to fix the unique solution we have to fix additional boundary data Γa Bdl = 4π c I a and in general I a = 0. The constants I a depend only on time and they can be viewed as fictitious currents[36] which intersect the loops Γ a . In the case of genus n = 0 (spherical) wormhole there is only one such a non-trivial loop which goes through the wormhole throat. In the case of genus n = 1 wormhole (doughnut -shaped throat) we have already two such loops, one goes through the throat and one additional goes through the hole in the center of the doughnut and surrounds the throat.
The first class produces the magnetic field of a wormhole which can be described by magnetic poles placed in two different entrances into the throat. The two entrances have opposite magnetic poles. If the distance between the entrances is big enough the resulting field is very weak and when crossing such a field high-energy charged particles only slightly change the direction of propagation. The field can be strong only very close to the throat entrances. However, since charged particles can freely propagate along the field lines (which are roughly orthogonal to entrances), the particles captured by the field are distributed in the whole region between the entrances. Such fields have the long-range character and can be used to explain the origin of long-correlated magnetic fields in voids [23,24] and, more generally, of primordial magnetic fields [25].
The situation changes when the wormhole possesses also the field of the second class. The second class corresponds to the field produced by a single loop of a current (the loop of the corresponding fictitious current goes inside of the surface of the doughnut). In this case the field lines of force repeat the shape of the entrance (the shape of a doughnut) which corresponds to the fields observed in spiral galaxies [31,32]. The most strong field is close to the entrance and particles captured by the field remain always near the entrance. In the primordial plasma before the recombination such a wormhole traps all baryons[37] propagating near it and thus forms a primeval structure of a galaxy (a protogalaxy). Such a scheme works only for wormholes which possess sufficiently strong magnetic fields, since high-energy baryons cannot be captured by the wormhole. The mean energy of baryons is determined by the temperature which depends on the redshift. The intensity of the magnetic field also depends on the redshift. While baryons are relativistic particles the threshold value of the fictitious (or equivalent) current does not depend on time.
Indeed, only particles below the threshold energy are captured by the wormhole magnetic field [12] which is given by where e is the electron charge and R w is the biggest radius of the doughnut -shaped throat of the wormhole. The energy of relativistic baryons behaves with the redshift as T = T γ (1 + z), where T γ is the present day temperature of CMB radiation. The intensity of the magnetic field can be estimated by the value in the hole of the doughnut. We take it as where κ = 2π in the center of the doughnut hole and κ = 2R w /r w close to the throat (surface of the doughnut) where the field reaches the maximum value. The field depends on the parameter I (which is the fictitious or equivalent current) and the big radius R w of the doughnut which also depends on the redshift as R w = R 0 /(1 + z).
The constant I behaves with the redshift as I = I 0 (1+z). Indeed the invariant characteristics is the number of magnetic lines captured by the throat (which go through the internal hole of the doughnut). This gives S Bds = const = Φ, where S an arbitrary surface whose boundary contour γ (dual to Γ) lays on the surface of the throat and cannot be contracted to a point, so that all magnetic lines intersect S only once. Taking the minimal surface S we find Φ ∼ 2π 2 c R w I which gives the behavior I ∼ 1/R w ∼ 1/a, where a = a 0 /(1 + z) is the scale factor of the Universe. We point out that the same dependence on the redshift z follows from the fact that the energy density of the magnetic field ρ B ∼ B 2 /4π decreases with the scale factor as ρ B ∼ 1/a 4 . This gives the threshold value for the equivalent current which defines the intensity of the magnetic field as where r γ = e 2 3kTγ . It is convenient to express r γ as follows r γ = r p (1 + z r ), where r p = e 2 mpc 2 is the classical radius of the proton and 1 + z r = mpc 2 3kTγ ∼ 10 12 is the redshift at which baryons become relativistic particles. It is curious that the threshold value I th is extremely small. It does not depend on the absolute size of the wormhole (which is given by the big radius R w ) but only on the ratio of the wormhole radii κ = 2R w /r w . All wormholes with the present day values I 0 > I th strongly interact with baryons. We may say that they are frozen into baryons and, therefore, peculiar motions of baryons repeat peculiar motions of such wormholes. Wormholes with smaller magnetic fields I 0 < I th slightly interact with baryons and can be considered as free objects. They may participate in independent from baryons motions. As we shall see on the early stage of the evolution of the Universe, before the recombination, they cannot capture baryons and therefore do not form an enhancement in the baryon density. On latter stages upon reheating they may capture charged particles and form cosmic rays and compact sources of synchrotron radiation. The relativistic stage for baryons finishes at the redshift z r upon which (e.g., for z < z r ) baryons become non-relativistic and the above consideration brakes.
At redshifts z r > z > z rec baryons are non-relativistic, while upon the recombination z rec baryons form neutral Hydrogen atoms and do not interact with magnetic fields of all wormholes. They again start to interact with wormholes only at the epoch of re-ionization when first stars have fired.
On the stage z r > z > z rec for every particular wormhole it is possible to define the critical redshift z 0 when it captures baryons (for z > z 0 ). The critical value z 0 can be estimated as follows. The mean energy of baryons is m p V 2 /2 = 3kT /2 with T = T γ (1 + z). Then the critical redshift can be found from the inequality r B = V ωB = V mpc eB < R w , where r B and ω B are Larmor radius and frequency respectively for protons. This gives or equivalently This defines the critical redshift z 0 at which the wormhole starts to trap baryons as (1 + z r ).
The relation between z 0 and I 0 can be rewritten as When z 0 = z r we get I 0 = I th and this gives the absolute threshold of the field. At redshifts z > z r we get into the epoch where baryons are relativistic particles and wormholes with I 0 < I th do not bound baryons at all. There is one more critical value I rec which corresponds to z 0 = z rec .
All fields with I 0 > I rec are strong enough to capture baryons during the whole evolution z > z rec .

III. THE NUMBER OF BARYONS IN TRAPS
The efficiency of wormhole traps can be described by the number of baryons collected. The number of baryons collected around magnetic wormholes depends on the proton diffusion length ℓ(z) and the two radii of a wormhole throat R w (z) and r w (z) ≪ R w . This number can be estimated as the increase of the effective volume of the torus-shaped throat where < n b > is the mean density of baryons and V = 2π 2 R w r 2 w is the throat volume. We define the parameter δ b = ∆N/(V < n b >) which depends on the position in space. Close to wormhole throats δ b > 0, while sufficiently far from the wormhole δ b < 0 since baryons from those regions have captured by the wormhole. The value < δ 2 b >= b, where brackets define the averaging over the space, relates to the baryon clumping factor b = (< n 2 b > − < n b > 2 )/ < n b > 2 , e.g., [26,27]. In the case ℓ(z) ≪ r w (z) we get In the intermediate case r w (z) ≪ ℓ(z) ≪ R w (z) we find the estimate And in the case ℓ(z) ≫ R w (z) the estimate has the order On the stage z > z r protons are relativistic particles, plasma is degenerate, and the length of the proton propagation can be estimated by the value of the horizon size ℓ(z) ∼ l h = c/H(z). At the redshift z = z r it is extremely small and has the order ℓ(z r ) ∼ (7 ÷ 8) × 10 −15 pc. Consider a wormhole throat with the big radius R w (0) ∼ 15kpc which corresponds to a galaxy size [5]. The small radius is not directly relates to the ring thickness in a galaxy. Nevertheless in expressions (4)-(6) we should use the thickness which gives r w ∼ 0.2R w . Then we find R w (z r ) ∼ 15 × 10 −9 pc ≫ ℓ(z r ) and from (4) we get δ b (z r ) ∼ (0.5 ÷ 0.6) × 10 −5 . Such a value is too small to form a galaxy without additional means (e.g., a dark matter clump).
At the recombination z rec = 1100 the proton diffusion length has the co-moving value of the order ℓ(z rec ) ∼ 0.4 ÷ 1pc, while R w (z rec ) ∼ 13.6pc and we still may consider ℓ(z rec ) ≪ R w and ℓ r w . Therefore we again may use (4) and find δ b (z) ∼ 0.32÷0.8. Such a big value shows that the respective protogalaxy forms immediately after the recombination and is in the non-linear regime. We should expect that wormholes with such strong clumps of baryons depart the Hubble expansion very soon and form rather small objects. Smaller wormholes form too strong inhomogeneities before recombination and probably collapse to blackholes. To be consistent with the present day size of a typical ring galaxy the wormhole radius should be at least two orders bigger R w (0) ∼ 1M pc.

IV. CONCLUSIONS
In conclusion we point out two important facts. First one is that the wormholes whose size R(z rec ) exceeds the value ℓ(z rec ) more than on the factor 10 3 do not form a sufficient enhancement in the baryon number density and therefore do not form primeval ring galaxies. The second fact is that upon the recombination z < z rec the doughnut -shaped wormholes do not interact with baryons and evolve. Therefore, they may leave the galaxy formed. They either expand or collapse forming a magnetized blackhole in the middle but the magnetic field may retain. If the wormhole expands, by the measuring the mean magnetic field in a ring galaxy one may predict the present day value of the big radius R = κI cB of such a wormhole and determine its present day position. This may be used in the direct search for wormhole traces in the Universe.
[34] To illustrate such a factorization one may consider flat space in which one of coordinates, say x, becomes periodic x = Rϕ/(2π) with the angle 0 < ϕ < 2π. Then the space becomes a cylinder with the radius R. In the open model (on space of a constant negative curvature) any parallel geodesics (ϕ = 0 and ϕ = 2π) diverge and the distance between them changes R(ℓ), where ℓ is a parameter along the geodesic line. Therefore, if we move along the geodesic line from the point where the distance is the shortest Rmin, the space opens out R → ∞ and becomes unrestricted. This is illustrated on Fig. 1. in Ref. [12]. In other words, the simplest stable wormhole corresponds simply to a torus on the Lobachevsky space.
[35] The simplest scheme to get a general genus -n wormhole can be described as follows. We take a couple of equal spheres with n handles in space, remove the inter-nal regions (insides of the spheres), and glue along their surfaces (the so-called Hegor diagrams). If we take two simple spheres without handles the resulting space corresponds to a spherical wormhole (throat is the sphere). The sphere with a handle is the torus. A couple of toruses corresponds to the doughnut shaped wormhole, etc.. [36] We point out that the currents are fictitious, for from the point of view of the Hegor diagrams they take place in inner regions of spheres which are removed, i.e., in fictitious regions.
[37] We present here estimates for baryons only, since leptons are much lighter than baryons and in the primordial plasma leptons simply follow baryons (bounded by the Coulomb potential).