aa r X i v : . [ g r- q c ] A ug August 28, 2018 0:36 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ws-procs961x669-ArbuzovaE page 1 Spontaneous and Gravitational Baryogenesis
E. V. Arbuzova ∗ Department of Higher Mathematics, Dubna State University ,Dubna, 141983, Moscow region, RussiaPhysics Department, Novosibirsk State University,Novosibirsk, 630090, Russia ∗ Some problems of spontaneous and gravitational baryogenesis are discussed. Gravitymodification due to the curvature dependent term in gravitational baryogensis scenariois considered. It is shown that the interaction of baryonic fields with the curvaturescalar leads to strong instability of the gravitational equations of motion and as a resultto noticeable distortion of the standard cosmology.
Keywords : cosmology, baryogenesis, gravitational equations, modified theories of gravity.
1. Introduction
Observations show that at least the region of the Universe around us is matter-dominated. Though we understand how the matter-antimatter asymmetry may becreated, the concrete mechanism is yet unknown. The amount of antimatter isvery small and it can be explained as the result of high energy collisions in space.The existence of large regions of antimatter in our neighbourhood would producehigh energy radiation as a consequence of matter-antimatter annihilation, which isnot observed. Any initial asymmetry at inflation could not solve the problem ofobserved excess of matter over antimatter, because the energy density associatedwith baryonic number would not allow for sufficiently long inflation.On the other hand, matter and antimatter seem to have similar properties andtherefore we could expect a matter-antimatter symmetric Universe. A satisfactorymodel of our Universe should be able to explain the origin of the local observedmatter-antimatter asymmetry. The term baryogenesis means the generation ofthe asymmetry between baryons (basically protons and neutrons) and antibaryons(antiprotons and antineutrons).In 1967 Andrey Sakharov pointed out 3 ingredients, today known as
Sakharovprinciples , to produce a matter-antimatter asymmetry from an initially symmetricUniverse. These conditions include: 1) non-conservation of baryonic number; 2)breaking of symmetry between particles and antiparticles; 3) deviation from thermalequilibrium. However, not all of three Sakharov principles are strictly necessary.In what follows we briefly discuss some features of spontaneous baryogenesis(SBG) and concentrate in more detail on gravitational baryogenesis (GBG). Boththese mechanisms do not demand an explicit C and CP violation and can pro-ceed in thermal equilibrium. Moreover, they are usually most efficient in thermal ugust 28, 2018 0:36 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ws-procs961x669-ArbuzovaE page 2 equilibrium.The statement that the cosmological baryon asymmetry can be created by spon-taneous baryogenesis in thermal equilibrium was mentioned in the original paperby Cohen and Kaplan and developed in subsequent papers , for review see .The term ”spontaneous” is related to spontaneous breaking of a global U (1)-symmetry, which ensures the conservation of the total baryonic number in the un-broken phase. This symmetry is supposed to be spontaneously broken and in thebroken phase the Lagrangian density acquires the additional term L SB = ( ∂ µ θ ) J µB , (1)where θ is the Goldstone field and J µB is the baryonic current of matter fields, whichbecomes non-conserved.For a spatially homogeneous field, θ = θ ( t ), the Lagrangian is reduced to thesimple form L SB = ˙ θ n B , n B ≡ J B , (2)where time component of a current is the baryonic number density of matter, so it istempting to identify ˙ θ with the chemical potential, µ B , of the corresponding system.However, such identification is questionable and depends upon the representationchosen for the fermionic fields . It is heavily based on the assumption ˙ θ ≈ const ,which is relaxed in the work . But still the scenario is operative and presents abeautiful possibility to create an excess of particles over antiparticles in the Universe.Subsequently the idea of gravitational baryogenesis (GBG) was put forward ,where the scenario of SBG was modified by the introduction of the coupling of thebaryonic current to the derivative of the curvature scalar R : L GBG = 1 M ( ∂ µ R ) J µB , (3)where M is a constant parameter with the dimension of mass.In the presented talk we demonstrate that the addition of the curvature de-pendent term (3) to the Hilbert-Einstein Lagrangian of General Relativity (GR)leads to higher order gravitational equations of motion, which are strongly unstablewith respect to small perturbations. The effects of this instability may drasticallydistort not only the usual cosmological history, but also the standard Newtoniangravitational dynamics. We discovered such instability for scalar baryons andfound similar effect for the more usual spin one-half baryons (quarks) .
2. Gravitational baryogenesis with scalar baryons
Let us start from the model where baryonic number is carried by scalar field φ withpotential U ( φ, φ ∗ ). An example with baryonic current of fermions will be consideredin the next section.The action of the scalar model has the form: A = Z d x √− g (cid:20) m P l π R + 1 M ( ∂ µ R ) J µ − g µν ∂ µ φ ∂ ν φ ∗ + U ( φ, φ ∗ ) (cid:21) − A m , (4) ugust 28, 2018 0:36 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ws-procs961x669-ArbuzovaE page 3 where m P l = 1 . · GeV is the Planck mass, A m is the matter action, J µ = g µν J ν , and g µν is the metric tensor of the background space-time. We assumethat initially the metric has the usual GR form and study the emergence of thecorrections due to the instability described below.In contrast to scalar electrodynamics, the baryonic current of scalars is notuniquely defined. In electrodynamics the form of the electric current is dictatedby the conditions of gauge invariance and current conservation, which demand theaddition to the current of the so called sea-gull term proportional to e A µ | φ | , where A µ is the electromagnetic potential.On the other hand, a local U (1)-symmetry is not imposed on the theory deter-mined by action (4). It is invariant only with respect to a U (1) transformationswith a constant phase. As a result, the baryonic current of scalars is considerablyless restricted. In particular, we can add to the current an analogue of the sea-gullterm, ∼ ( ∂ µ R ) | φ | , with an arbitrary coefficient.In our paper we study the following two extreme possibilities, when the sea-gullterm is absent and the current is not conserved, or the sea-gull term is included withthe coefficient ensuring current conservation. In both cases no baryon asymmetrycan be generated without additional interactions. It is trivially true in the secondcase, when the current is conserved, but it is also true in the first case despite thecurrent non-conservation, simply because the non-zero divergence D µ J µ does notchange the baryonic number of φ but only leads to redistribution of particles φ in thephase space. So to create any non-zero baryon asymmetry we have to introduce aninteraction of φ with other particles which breaks conservation of B by making thepotential U non-invariant with respect to the phase rotations of φ , as it is describedbelow.If the potential U ( φ ) is not invariant with respect to the U (1)-rotation, φ → exp ( iβ ) φ , the baryonic current defined in the usual way J µ = iq ( φ ∗ ∂ µ φ − φ∂ µ φ ∗ ) (5)is not conserved. Here q is the baryonic number of φ and we omitted index B incurrent J µ .With this current and Lagrangian (4) the equation for the curvature scalar, R ,takes the form: m P l π R + 1 M (cid:2) ( R + 3 D ) D α J α + J α D α R (cid:3) − D α φ D α φ ∗ + 2 U ( φ ) = − T µµ , (6)where D µ is the covariant derivative in metric g µν (of course, for scalars D µ = ∂ µ )and T µν is the energy-momentum tensor of matter obtained from action A m .According to definition (5), the current divergence is: D µ J µ = 2 q M (cid:2) D µ R ( φ ∗ D µ φ + φD µ φ ∗ ) + | φ | D R (cid:3) + iq (cid:18) φ ∂U∂φ − φ ∗ ∂U∂φ ∗ (cid:19) . (7)If the potential of U is invariant with respect to the phase rotation of φ , i.e. U = U ( | φ | ), the last term in this expression disappears. Still the current remains ugust 28, 2018 0:36 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ws-procs961x669-ArbuzovaE page 4 non-conserved, but this non-conservation does not lead to any cosmological baryonasymmetry. Indeed, the current non-conservation is proportional to the product φ ∗ φ , so it can produce or annihilate an equal number of baryons and antibaryons.To create cosmological baryon asymmetry we need to introduce new types ofinteractions, for example, the term in the potential of the form: U = λ φ + λ ∗ φ ∗ . This potential is surely non invariant w.r.t. the phase rotation of φ and caninduce the B-non-conserving process of transition of two scalar baryons into twoantibaryons, 2 φ → φ .Let us consider solution of the above equation of motion in cosmology. Themetric of the spatially flat cosmological FRW background can be taken as: ds = dt − a ( t ) d r . (8)In the homogeneous case the equation for the curvature scalar (6) takes theform: m P l π R + 1 M h ( R + 3 ∂ t + 9 H∂ t ) D α J α + ˙ R J i = − T ( tot ) , (9)where J is the baryonic number density of the φ -field, H = ˙ a/a is the Hubbleparameter, and T ( tot ) is the trace of the energy-momentum tensor of matter includ-ing contribution from the φ -field. In the homogeneous and isotropic cosmologicalplasma T ( tot ) = ρ − P , (10)where ρ and P are respectively the energy density and the pressure of the plasma.For relativistic plasma ρ = π g ∗ T /
30 with T and g ∗ being the plasma tempera-ture and the number of particle species in the plasma. The Hubble parameter isexpressed through ρ as H = 8 πρ/ (3 m P l ) ∼ T /m P l .The covariant divergence of the current is given by the expression (7). In thehomogeneous case we are considering it takes the form: D α J α = 2 q M h ˙ R ( φ ∗ ˙ φ + φ ˙ φ ∗ ) + ( ¨ R + 3 H ˙ R ) φ ∗ φ i + iq (cid:18) φ ∂U∂φ − φ ∗ ∂U∂φ ∗ (cid:19) . (11)To derive the equation of motion for the classical field R in the cosmologicalplasma we have to take the expectation values of the products of the quantumoperators φ , φ ∗ , and their derivatives. Performing the thermal averaging, we find h φ ∗ φ i = T , h φ ∗ ˙ φ + ˙ φ ∗ φ i = 0 . (12)Substituting these average values into Eq. (9) and neglecting the last term inEq. (11) we obtain the fourth order differential equation: m P l π R + q M (cid:0) R + 3 ∂ t + 9 H∂ t (cid:1) h(cid:16) ¨ R + 3 H ˙ R (cid:17) T i + 1 M ˙ R h J i = − T ( tot ) . (13)Here h J i is the thermal average value of the baryonic number density of φ . It isassumed to be zero initially and generated as a result of GBG. We neglect this term, ugust 28, 2018 0:36 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ws-procs961x669-ArbuzovaE page 5 since it is surely small initially and probably subdominant later. Anyhow it doesnot noticeably change the exponential rise of R at the onset of the instability.Eq. (13) can be further simplified if the variation of R ( t ) is much faster than theuniverse expansion rate or in other words ¨ R/ ˙ R ≫ H . Correspondingly the temper-ature may be considered adiabatically constant. The validity of these assumptionis justified a posteriori after we find the solution for R ( t ).Keeping only the linear in R terms and neglecting higher powers of R , such as R or HR , we obtain the linear differential equation of the fourth order: d Rdt + µ R = − T ( tot ) , where µ = m P l M πq T . (14)The homogeneous part of this equation has exponential solutions R ∼ exp( λt )with λ = | µ | exp ( iπ/ iπn/ , (15)where n = 0 , , , λ . This indicates that thecurvature scalar is exponentially unstable with respect to small perturbations, so R should rise exponentially fast with time and quickly oscillate around this risingfunction.Now we need to check if the characteristic rate of the perturbation explosion isindeed much larger than the rate of the universe expansion, that is:( Re λ ) > H = (cid:18) πρ m P l (cid:19) = 16 π g ∗ T m P l , (16)where ρ = π g ∗ T /
30 is the energy density of the primeval plasma at temperature T and g ∗ ∼ −
100 is the number of relativistic degrees of freedom in the plasma.This condition is fulfilled if 20252 π q g ∗ m P l M T > , (17)or, roughly speaking, if T ≤ m / P l M / . Let us stress that at these temperaturesthe instability is quickly developed and the standard cosmology would be destroyed.If we want to preserve the successful big bang nucleosynthesis (BBN) resultsand impose the condition that the development of the instability was longer thanthe Hubble time at the BBN epoch at T ∼ M should be extremelysmall, M < − MeV. The desire to keep the standard cosmology at smaller T would demand even tinier M . A tiny M leads to a huge strength of coupling (3).It surely would lead to pronounced effects in stellar physics. ugust 28, 2018 0:36 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ws-procs961x669-ArbuzovaE page 6
3. Gravitational baryogenesis with fermions
Let us now generalize results, obtained for scalar baryons, to realistic fermions. Westart from the action in the form A = Z d x √− g (cid:20) m P l π R − L m (cid:21) (18)with L m = i Qγ µ ∇ µ Q − ∇ µ ¯ Q γ µ Q ) − m Q ¯ Q Q + i Lγ µ ∇ µ L − ∇ µ ¯ Lγ µ L ) − m L ¯ L L (19)+ gm X (cid:2) ( ¯ Q Q c )( ¯ QL ) + ( ¯ Q c Q )( ¯ LQ ) (cid:3) + fm ( ∂ µ R ) J µ + L other , where Q is the quark (or quark-like) field with non-zero baryonic number, L isanother fermionic field (lepton), ∇ µ is the covariant derivative of Dirac fermionin tetrad formalism. m is a constant parameter with dimension of mass and f is dimensionless coupling constant which is introduced to allow for an arbitrarysign of the curvature dependent term in the above expression. J µ = ¯ Qγ µ Q is thequark current with γ µ being the curved space gamma-matrices, L other describes allother forms of matter. The four-fermion interaction between quarks and leptons isintroduced to ensure the necessary non-conservation of the baryon number with m X being a constant parameter with dimension of mass and g being a dimensionlesscoupling constant. In grand unified theories m X may be of the order of 10 − GeV.Varying the action (18) over metric, g µν , and taking trace with respect to µ and ν , we obtain the following equation of motion for the curvature scalar: − m P l π R = m Q ¯ QQ + m L ¯ LL + 2 gm X (cid:2) ( ¯ Q Q c )( ¯ QL ) + ( ¯ Q c Q )( ¯ LQ ) (cid:3) − fm ( R + 3 D ) D α J α + T other , (20)where T other is the trace of the energy momentum tensor of all other fields. Atrelativistic stage, when masses are negligible, we can take T matter = 0. The averageexpectation value of the interaction term proportional to g is also small, so thecontribution of all matter fields may be neglected.As we see in what follows, kinetic equation leads to an explicit dependence on R of the current divergence, D α J α , if the current is not conserved. As a result weobtain 4th order equation for R .As previously, we study solutions of Eq. (20) in cosmology in homogeneous andisotropic FRW background with the metric ds = dt − a ( t ) d r . The curvature isa function only of time and the covariant derivative acting on a vector V α , whichdepends only on time and has only time component, has the form: D α V α = ( ∂ t + 3 H ) V t , (21) ugust 28, 2018 0:36 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ws-procs961x669-ArbuzovaE page 7 where H = ˙ a/a is the Hubble parameter.As an example let us consider the reaction q + q ↔ ¯ q + l , where q and q are quarks with momenta q and q , while ¯ q and l are antiquark and lepton withmomenta q and l . We use the same notations for the particle symbol and for theparticle momentum. The kinetic equation for the variation of the baryonic numberdensity n B ≡ J t through this reaction in the FRW background has the form:( ∂ t + 3 H ) n B = I collB , (22)where the collision integral for space and time independent interaction is equal to: I collB = − B q (2 π ) Z dν q ,q dν ¯ q ,l δ ( q + q − q − l ) (cid:2) | A ( q + q → ¯ q + l ) | f q f q − | A (¯ q + l → q + q ) | f ¯ q f l (cid:3) , (23)where A ( a → b ) is the amplitude of the transition from state a to state b , B q isthe baryonic number of quark, f a is the phase space distribution (the occupationnumber), and dν q ,q = d q E q (2 π ) d q E q (2 π ) , (24)where E q = p q + m is the energy of particle with three-momentum q and mass m . The element of phase space of final particles, dν ¯ q ,l , is defined analogously.We neglect the Fermi suppression factors and the effects of gravity in the collisionintegral. This is generally a good approximation.The calculations are strongly simplified if quarks and leptons are in equilibriumwith respect to elastic scattering and annihilation. In this case their distributionfunctions take the form f = 1 e ( E/T − ξ ) + 1 ≈ e − E/T + ξ , (25)with ξ = µ/T being dimensionless chemical potential, different for quarks, ξ q , andleptons, ξ l .The assumption of kinetic equilibrium is well justified since it is usually enforcedby very efficient elastic scattering. Equilibrium with respect to annihilation, say,into two channels: 2 γ and 3 γ , implies the usual relation between chemical potentialsof particles and antiparticles, ¯ µ = − µ .The baryonic number density is given by the expression: n B = Z d q E q (2 π ) ( f q − f ¯ q )= g S B q (cid:18) µT + µ π (cid:19) = g S B q T (cid:18) ξ + ξ π (cid:19) , (26)where T is the cosmological plasma temperature, g S and B q are respectively thenumber of the spin states and the baryonic number of quarks. ugust 28, 2018 0:36 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ws-procs961x669-ArbuzovaE page 8 We can use another representation of the quark field: Q = exp( if R/m ) Q (27)analogously to what is done in our paper . Written in terms of Q Lagrangian(20) would not contain terms proportional to f /m , but dependence on such termswould reappear in the interaction term as:2 gm X h e − ifR/m ( ¯ Q Q c )( ¯ Q L ) + e ifR/m ( ¯ Q c Q )( ¯ LQ ) i . (28)Nevertheless we obtain the same fourth order equation for the evolution of curvature,as for non-rotated field Q .Since the transition amplitudes, which enter the collision integral, are obtainedby integration over time of the Lagrangian operator (28), taken between the initialand final states, the energy conservation delta-function in Eq. (23) would be modi-fied due to time dependent factors exp[ ± if R ( t ) /m ]. In the simplest case, which isusually considered in gravitational (and spontaneous) baryogenesis, a slowly chang-ing ˙ R is taken, so we can approximate R ( t ) ≈ ˙ R ( t ) t . In this case the energy is notconserved but the energy conservation condition is trivially modified, as δ [ E ( q ) + E ( q ) − E ( q ) − E ( l )] →→ δ [ E ( q ) + E ( q ) − E ( q ) − E ( l ) − f ˙ R ( t ) /m ] . (29)Thus the energy is non-conserved due to the action of the external field R ( t ). Delta-function (29) is not precise, but the result is pretty close to it, if ˙ R ( t ) changes verylittle during the effective time of the relevant reactions.If the dimensionless chemical potentials ξ q and ξ l , as well as f ˙ R ( t ) /m /T , aresmall, the collision integral can be written as: I collB ≈ C I g T m X " f ˙ R ( t ) m T − ξ q + ξ l , (30)where C I is a positive dimensionless constant. The factor T appears for reactionswith massless particles and the power eight is found from dimensional consideration.Because of conservation of the sum of baryonic and leptonic numbers ξ l = − ξ q / R ( t ) is analogous to fast variation of ˙ θ ( t )studied in our paper . Clearly, it is much more complicated technically. Here weconsider only the simple situation with quasi-stationary background and postponemore realistic time dependence of R ( t ) for the future work.For small chemical potential the baryonic number density (26) is equal to n B ≈ g s B q ξ q T , (31)and if the temperature adiabatically decreases in the course of the cosmologicalexpansion, according to ˙ T = − HT , equation (22) turns into˙ ξ q = Γ " f ˙ R ( t )10 m T − ξ q , (32) ugust 28, 2018 0:36 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ws-procs961x669-ArbuzovaE page 9 where Γ ∼ g T /m X is the rate of B-nonconserving reactions.If Γ is in a certain sense large, this equation can be solved in stationary pointapproximation as ξ q = ξ eqq − ˙ ξ eqq / Γ , where ξ eqq = 910 f ˙ Rm T . (33)If we substitute ξ eqq into Eq. (20) we arrive to the fourth order equation for R .According to the comment below Eq. (20), the contribution of thermal matterinto this equation can be neglected, and we arrive to the very simple fourth orderdifferential equation: d Rdt = λ R, (34)where λ = C λ m P l m /T with C λ = 5 / (36 πf g s B q ). Deriving this equation weneglected the Hubble parameter factor in comparison with time derivatives of R . Itis justified a posteriori because the calculated λ is much larger than H .Evidently equation (34) has extremely unstable solution with instability time byfar shorter than the cosmological time. This instability would lead to an explosiverise of R , which may possibly be terminated by the nonlinear terms proportionalto the product of H to lower derivatives of R . Correspondingly one may expectstabilization when HR ∼ ˙ R , i.e. H ∼ λ . Since˙ H + 2 H = − R/ , (35) H would also exponentially rise together with R , H ∼ exp( λt ) and λH ∼ R . Thusstabilization may take place at R ∼ λ ∼ m P l m /T . This result should be comparedwith the normal General Relativity value R GR ∼ T matter /m P l , where T matter is thetrace of the energy-momentum tensor of matter.
4. Discussion and conclusion
For more accurate analysis numerical solution will be helpful, which we will per-form in another work. The problem is complicated because the assumption of slowvariation of ˙ R quickly becomes broken and the collision integral in time dependentbackground is not so simply tractable as the usual stationary one. The techniquefor treating kinetic equation in non-stationary background is presented in Ref. .For evaluation of R ( t ) in this case numerical calculations are necessary, which willbe presented elsewhere. Here we describe only the basic features of the new effectof instability in gravitational baryogenesis.To conclude we have shown that gravitational baryogenesis in the simplest ver-sions discussed in the literature is not realistic because the instability of the emerginggravitational equations destroys the standard cosmology. Some stabilization mech-anism is strongly desirable. Probably stabilization may be achieved in a version of F ( R )-theory. ugust 28, 2018 0:36 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ws-procs961x669-ArbuzovaE page 10 Acknowledgement
This work was supported by the RSF Grant N 16-12-10037. The author ex-presses sincere gratitude to Harald Fritzsch for his invitation and for the opportu-nity to present the talk at the Conference on Particles and Cosmology. She wouldlike to thank Kok Khoo Phua for his kind hospitality at NTU, Singapore.
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