Baryon number of the Universe as a result of extra space dynamics
aa r X i v : . [ g r- q c ] D ec Baryon number of the Universe as a resultof extra space dynamics
A.V. Grobov ∗ S. G. Rubin † National Research Nuclear University “MEPhI”
Аннотация
Origin of baryon asymmetry is studied in the framework of extradimensional approach. Baryon excess production and the symmetrizationof extra-space are performed simultaneously. Baryon number is conservedlong after the inflationary stage when the U(1) symmetry is achieved.
Prediction of antimatter made by Dirac in 1928 and discovery of positronsin cosmic rays initiated the study of antiparticles. Now it is firmly establishedthat our Universe mainly consists of particles (baryon matter). Antiparticles(antibaryons) form only a small fraction. The number density of baryonsis characterized by the ratio: ∆ B = ∆ n B s = 0 . · − (1)where s - is the entropy density and ∆ n B - is the baryon excess. Thisquantity is constant if the baryon number is conserved and the Universeexpands adiabatically.Origin of the baryon asymmetry is one of the key problem in cosmology.The question is whether the Universe was born asymmetric or was theasymmetry formed during the evolution of the Universe? Why the baryoncharge is conserved today with high accuracy whereas the baryon excesswas created in the past at some (also unclear) stage of the Universeevolution [1], [2]?First attempt to explain the phenomenon of baryon asymmetry wasdone by Sakharov in 1967. According to his approach there are threeknown conditions that had to be fulfilled in order to originate baryonasymmetry at some stage. Nowadays there are a great variety of differentapproaches [3], [4], [5], [7]. Up to now there is no unique preferable model. ∗ [email protected] † [email protected] s a result there is still no proper answer to the question concerning theemergence of the baryon asymmetry. On the other hand, the idea of extraspace almost inevitably leads to charge non conservation. Indeed in theframework of multidimensional gravity observed low energy symmetriesare the consequences of isometries of extra space [11]. As was discussedin [12, 13] there are no any Killing vectors of a nucleated manifold in thevery beginning. The first stage of an extra space formation consists ofits symmetrization. Hence, there are no conserved charges at this stage.Charge conservation appears much later when the extra space geometryacquires appropriate Killing vectors. The situation is similar to thoseconcerning the problem of baryon asymmetry.In the present paper we consider a mechanism of the baryon asymmetrygeneration accompanied by symmetrization of extra space. It is assumedthat the baryon symmetry (asymmetry) is the consequence of symmetry(asymmetry) of extra space. The corresponding symmetry of the theoryis restored during the process of evolution at later stages.Another approaches are known as well. The baryon production dueto dilaton stabilization is discussed in [21]. Brane idea could be fruitfulin this context, see [6] where non conservation of global charge during abrane nucleation was considered.In general models based on large and universal extra dimensions havea serious problem. Indeed, if the true gravity scale is in the TeV range,early stages of the Universe formation would be seriously modified [7]. Inparticular it could influence a small reheating temperature, the amplitudefluctuation of inflaton, the electroweak symmetry breaking and the particleproduction during the reheating stage. One of the way to cure the problemis to take into account a time variation of the gravitational constant. Thelatter should freeze before the nucleogenesis [8, 9, 10]. There is no suchproblem if an extra space is small enough so that its influence on theinflationary dynamics could be neglected. Consider a D = 8 -dimensional Riemannian manifold V = M × V × H with a metric G . The FRW metric with the scale factor a ( t ) of our 4-dimspace M is denoted as g µν ( x ) where x µ ( µ = 1 , , , are its coordinates.The subspace V with the topology T × T possesses a metric G ( V ) ab andis described by coordinates y a , a = 5 , in the interval ≤ y a < π .Hyperbolic subspace H with a radius r d and coordinates θ, φ plays asubsidiary role. The interval is chosen in the form ds = dt − dx − dy − dz − ( r c + h ab ) dy a +2 e a dy a dθ − r d dθ − r d sinh ( θ ) dφ (2)As was shortly discussed in the Introduction nucleation of manifoldscontaining a symmetry has zero probability [13]. In this connection considerthe metric of the subspace V being slightly deviated from symmetricalone. More definitely, suppose the following form of the metric G ( V ) ab = G ( V,stat ) ab + h ab ( t, y , y ) (3) ts stationary part G ( V,stat ) ab = diag ( r c , r c ) is invariant under the SO (2) transformations. The extra space V acquires this symmetry at late timeswhen the fluctuations h ab ( t, y , y ) decay into lighter particles. Restorationof the symmetry of our metric gives rise to the baryon charge conservation.The off-diagonal components of the metric are also important in thefollowing consideration. G a metric components are transformed underthe fundamental representation of the SO (2) (or U (1) ) group introducedabove. We suppose that the corresponding charge Q = Q B is the baryonicone and hence the field e a ≡ G a could form a baryonic condensate. Itsinteraction with quarks and leptons will be shortly discussed later in thesame way as it was done in [7].As will be shown later the baryon asymmetry takes place if h ab ( t, y , y ) = h ab ( t, y , y ) . We choose the simplest form - first term in the Fourier series h ab ( t, y , y ) = δ ab h ( t ) cos( y ) (4)to perform analytical estimation. The G a components though small butimportant. All other components of the metric G also contribute to theeffective low energy action. Nevertheless we will omit them since theydo not contribute to the baryon excess. It strongly facilitates an analysisleaving the idea untouched.The model is specified by the nonlinear action S = m D − D Z d D X √ G (cid:2) R + cR (cid:3) (5)There are two parameters in the model - m D and c while the metric tensorcontains another two - r c and r d . According to modern experiment theseextra space sizes must be smaller than ∼ − cm. Another restrictionfollowed from the fact that quantum fluctuations of a metric becomeimportant at m D scale. The inequalities /m D ≪ r c , r d . . TeV − (6)permit us to deal with classical behavior of the metric of small enoughextra space.The Ricci scalar R is a complicated function of the fields h ( t, y ) and e a ( t ) ≡ G a . Keeping in mind expression (4) the Ricci scalar can bewritten explicitly: R = R + R H + R h + R e + R he (7)where R - is a curvature of the 4-dimensional space-time g µν , R H is thecurvature of the H space, R h = 32 r c cos ( y )( ∂ t h ( t )) − r c sin ( y ) h ( t ) R e = 12 r c r d ( ∂ t e a ) − r d r c e a R he = − coth( θ ) r d r c sin( y ) e h ( t )( e + e ) . To simplify the analysis we will suppose that r d ≫ | c | . Effective Lagrangian
After integrating out the internal coordinates in expression (5) one obtainsthe effective Lagrangian S eff = Z d x √− g L eff , L eff = 12 ( ∂ t χ ) − m χ χ + 12 ( ∂ t ϕ ) − m ϕ ϕ + 12 ( ∂ t ϕ ) − m ϕ ϕ + c · λ ∗ χ ( ϕ + ϕ ) ϕ where field normalization χ = r d r c q π m D V θ h (8) ϕ a = r d r c q π m D V θ e a (9) V θ = Z φ + φ − Z θ + θ − sinh( θ )d θ d φ (10)was performed to obtain the standard form of the Lagrangian. Here weomitted terms describing the Einstein-Hilbert action and Λ − term becausethey do not play significant role in the model. To obtain them accurately,one should take into account all effects what is far from our purpose.Besides we deleted all interaction terms except the last one which isresponsible for the baryogenesis. It can be done if the metric fluctuationsare small h ( t, y , y ) ≪ r c , e a ≪ r c , r d . (11)The masses of the new fields are expressed in terms of initial parameters m χ = 13 r c , m ϕ = 4 r d (12)The expression for coupling constant λ ∗ = W θ /V θ π m D r c ( r d − c ) (13)contains the integral W θ = Z φ + φ − Z θ + θ − cosh( θ ) coth( θ )d θ d φ (14)over the space H that are worth discussing.The limits of integration are nontrivial due to complicated boundaryof compact hyperbolic manifolds. The boundary of our compact space H with the metric ds = dθ + sinh ( θ ) dφ (15)is described in the following way φ − = 0 , φ + = π + = 2 arth (cid:18) A · ctg ( π cos ( φ − π − r A cos ( φ − π − sin ( φ − π (cid:19) where A = 2 sin ( π ) . The lower limit θ − is worth discussing. From apure geometrical point of view θ − = 0 . On the other hand a classicaldescription is valid if size of the space is much greater than the Planckscale. Smaller regions can not be considered accurately due to strongquantum fluctuations inside them. So the scale of lower limit θ − of actionintegral is proportional to /M Planck . Usually, such a small value does notinfluence any effects. In our case the quantum fluctuations of the metricbecame important at the scale /m D instead of /M Planck . So the valueof the low limit of the integral over the angle θ should be θ − ≃ / ( r d m D ) .After integration V θ ≈ . , and W θ ≃ ln ( m D r d ) .In terms of complex field φ = ϕ + iϕ √ the effective Lagrangian has the form (we assume m ϕ ≡ m φ ): L eff = 12 ( ∂ t χ ) − m χ χ + ∂ t φ∂ t φ ∗ − m φ φφ ∗ (16) − λχ φ φ ∗ [ φ + φ ∗ ] where λ = − c · λ ∗ > and c < . The last term breaks global U (1) symmetry and therefore is responsible for asymmetrical baryosynthesis.In the modern epoch χ ( t → ∞ ) → so that the U (1) is restored. Thelatter group is isomorphic to the SO (2) group - exact symmetry group ofunperturbed metric G V in (3). Thus the process of symmetrization of theextra space V is accompanied by the baryon excess production.The final form of the effective action S r,ϑ = Z dta ( t ) (cid:20)
12 ( ∂ t χ ) − m χ χ + 12 ˙ r + 12 r ˙ ϑ − V ( r, ϑ, χ ) (cid:21) (17)contains the potential V ( r, ϑ, χ ) = m φ r λχ r cos ( ϑ ) Here the field φ is represented in the form φ ( t ) = r ( t ) e iϑ ( t ) / √ Effective Lagrangian (16) is similar to those considered in the frameworkof Affleck-Dine model [22]. The only essential difference is the presenceof additional field χ . The baryosynthesis is terminated when this fieldreaches zero value. In the FRW space the equations of motion for the fields χ, r and ϑ ¨ χ + 3 H ˙ χ + m χ χ = − λχr cos ( ϑ )¨ r + 3 H ˙ r − r ˙ ϑ + m φ r = − λχ r cos ( ϑ ) r ¨ ϑ + 3 Hr ˙ ϑ + 2 r ˙ ϑ ˙ r = λχ r sin(2 ϑ ) (18) ollow from action (17).Oscillations of the field ϑ are responsible for the generation of thebaryon excess. Indeed the dynamical equation for the field ϑ can be writtenin more suitable form a ∂∂t (cid:16) a r ˙ ϑ (cid:17) = − ∂V∂ϑ (19)The field φ is transformed under the fundamental representation of thegroup U (1) . The baryon charge connected to this group is calculated instandard manner n B = j = i ( φ ∗ ∂ φ − φ∂ φ ∗ ) = r ˙ ϑ . Substituting thisinto equation (19) we obtain the equation for baryon density a − ∂∂t ( a n B ) = − ∂V∂ϑ = λχ r sin (2 ϑ ) (20)with formal solution n B ( t ) = a ( t ) − λ Z tt in a ( t ′ ) r ( t ′ ) χ ( t ′ ) sin (2 ϑ ( t ′ )) dt ′ . (21)Our aim is to study the ability of the model to explain the observablebaryon density. According to [16] it can be done in quite simple andelegant way. Suppose that the dynamic of the field ϑ ruled by equation(19) elaborates baryon excess during one e-fold. Then the estimation ofintegral (21) gives n B ( t B ) ≃ e − λχ sin (2 ϑ ) r H − . (22)A baryon charge at the moment t B = H − of its creation is connected tothe modern baryon excess n B ( t ) n B ( t B ) = ( a ( t B ) /a ( t )) n B ( t ) = ( t H ) n B ( t ) . (23)Here we suppose the approximate equality a ( t B ) /a ( t ) ≃ ( t /t B ) / ≃ ( t H ) / to simplify the estimation.In this paper we suppose the masses of the field χ and r (12) is of order GeV for chosen set of the parameters, see the text above expression(28), what is much greater than the Hubble parameter in the consideredstage. It means that these fields quickly oscillate around zero. The periodsof these oscillations are much smaller than the period of oscillations of thefield ϑ what allows us to use average values like χ → h χ i in expression(22). Keeping in mind expression (23) we obtain the connection betweenthe parameters n B ( t ) = t − H − e − λ h χ ih r i sin (2 ϑ ) (24)The observed parameters included in this equation are t = 14 · sec =6 . · GeV − , n B = 2 . · − cm − = 1 . · − GeV . Finally,the relation between unknown parameters acquires the form (we suppose sin (2 ϑ ) ≈ ) λ h χ ih r i H ≈ . · GeV (25) ffective baryon production starts when a slow rolling of the field ϑ isterminated what leads to an additional connection H ≃ p λ h χ ih r i (26)followed from the third equation in (18). The estimations could be violatedat high energies by forth order terms which have been omitted in action(17). We assume that the process of baryon formation is most effective atmoderate energies where the quadratic terms dominate. At high energiesand large field values quick expansion of the Universe strongly reduces thebaryon number density. With estimation of the integral W θ ≈ ln ( m D r d ) λ can be written as λ ≈ − . · − cm D r c r d ln ( m D r d ) (27)Expressions (25) and (26) impose main restrictions on the parameters ofthe model.Let us specify parameters of the model and choose m D = 10 GeV, c = − m − D , r c = r d = 10 m − D . Initial values of the fields h and e a shouldbe small compared to the size of extra space (11) and we assume (8) h| χ |i ∼ h| φ |i ∼ m D . This set of parameters satisfies main equations(25) and (26) if the Hubble parameter H ∼ . · GeV. The Hubbleparameter lays in a wide range . GeV < H < GeV (28)if the process takes place after the inflation and before the primordialnucleosynthesis and our choice seems reasonable.In addition, there are several natural relations between the parameterslike those represented in (6) and (11). It can be easily checked that all ofthem are satisfied for chosen values of the parameters. The energy densityof the fields φ and χ is much smaller than the energy density of the inflatonfield. Indeed ρ χ ≃ m χ χ ∼ · m D = 10 − M pl . (29)At the same time the inflaton energy density is about ρ inf = 10 − M pl in the framework of the chaotic inflation.The relations mentioned above constrain the permissible range of parametersthat is represented in Fig. 1. One can see that there is substantial roomfor the parameters inside the triangle. Typically, they could vary in orderof magnitude so that there is no fine tuning in this model.In our approach we suppose that the observed symmetries are theresult of the appropriate symmetries of an extra space. It is known that theinternal SU (2) × U (1) electroweak symmetry of the Standard Model couldexist due to some symmetry of an extra space [23], [24]. The conservationof the baryon and lepton charges is supposed to be the result of some U (1) symmetries of extra space as well. In this paper we discuss the U (1) B symmetry of the extra space V which is responsible for the baryon chargeconservation. Those fermions transformed under fundamental representationof the U (1) B group are observed as baryons. The baryon charge stored by m D and r c . Parameters used in the text are marked as the smallsquare. the field φ should be transferred to matter fields. As was shown in [1, 20]it can be done due to the interaction term of the form L int = gφ ¯ QL + h.c., where Q is a wave function of some hypothetical heavy quark, L is a wavefunction of lepton and g is a Yukawa coupling constant.The coupling constant g should be quite small for not to destroy themodel developed above. Otherwise, quanta of the field φ would havedecayed into fermions before its condensate starts to oscillate after themoment t osc ∼ H − . The condensate of the field φ is evaporated duringthe time t decay ∼ Γ − φ where Γ φ is the decay probability. As was discussedin [25] its form is Γ φ = 4 π − / g m φ π (cid:18) m φ g h| φ |i (cid:19) / (30)if g h| φ |i > m φ . This inequality is true for the parameter values chosenabove. As the result the inequality t osc < t decay (31)gives the upper limit to the fermion coupling constant g ≤ . · − .Accordingly, upper limit for effective coupling constant g eff ≡ g / (4 π ) isapproximately . · − . This limit looks reasonable.If the symmetry groups discussed above are local ones the modelshould contain gauge fields. Gauge fields associated with baryon/leptoncharge are discussed in a literature, see e.g. [26]. In our case gauge fields hould be represented by off diagonal components of metric tensor (2).Nonobservation of baryon gauge field could be explained by a large massof its quanta. This subject worths separate paper. Among others there are two key problems - the observable baryon excess inthe Universe and the existence of various symmetries. The latter is tightlyconnected to appropriate symmetries of an extra space in the framework ofextra dimensional approach. So the second problem may be reformulatedas the question "Why an extra space possesses any symmetries?" . If itwas born having an arbitrary geometry, there must exist some periodduring which the symmetries are established. This means that no chargesare conserved during this stage. In the following stage the extra space isstabilized and acquires some symmetries. This leads to charges conservationin the modern epoch.The picture described above fits very well with the baryon problem.Non conservation of the baryon charge results in the baryon excess in thebeginning. In the modern epoch an extra space is supposed to be stableand possesses U (1) symmetry which relates to conservation of baryoncharge. In this paper we show that this mechanism is able to explain theobservable baryon excess. S.R. was supported by the Ministry of education and science of RussianFederation, project 14.A18.21.0789. The work of A.G. was supported byThe Ministry of education and science of Russian Federation, project14.132.21.1446. The authors are grateful to A. A. Korotkevich for clarifyingthe problem concerning a boundary of compact hyperbolic space.
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