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Home , Baryogenesis, Reionization

Impact of gravitational on cosmological evolution

Elena Arbuzova Dubna State University & Novosibirsk State University based on common works with A.D. Dolgov Phys.Rev. D94 (2016) 123501, arXiv:1607.01247 ( + V. A. Novikov) Phys.Lett. B769 (2017) 171-175; arXiv:1612.06206 JCAP 1706 (2017) 001; arXiv:1702.07477

QUARKS – 2018 XXth International Seminar on High Physics 27 May 2018 – 2 June 2018, Valday, Russia E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 1 / 27 Outline

Introduction Basics of Spontaneous Baryogenesis Gravitational Baryogenesis with Gravitational Baryogenesis with fermions Conclusions

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 2 / 27 Baryogenesis dominance in the : The amount of is very 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 produce high energy radiation as a consequence of matter-antimatter , which is not observed. Any initial at inflation could not solve the problem of observed excess of matter over antimatter, because the energy density associated with baryonic number would not allow for sufficiently long inflation. On the other hand, matter and antimatter seem to have similar properties and therefore we could expect a matter-antimatter symmetric universe. A satisfactory model of our Universe should be able to explain the origin of the matter-antimatter asymmetry. The term baryogenesis is used to indicate the generation of the asymmetry between and antibaryons.

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 3 / 27 History of the Universe Age Event 10−43 s 1019 GeV era (?) ? ? Inflation ? ? ?? Baryogenesis 10−11 s 200 GeV Electroweak symmetry breaking 10−5 s ∼ 100 MeV QCD 1 s – 15 min 0.05-1 MeV Big Bang 60 kyr 1 eV Matter-radiation equality 370 kyr 0.3 eV Recombination and 0.2 – 1 Gyr 15 – 50 eV 1 – 10 Gyr 3 – 15 eV 6 Gyr 4 K Transition to an accelerating Universe 9 Gyr 3 K Formation of the Solar System 13.8 Gyr 2.7 K Today

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 4 / 27 Sakharov Principles (1967)

1 Non-conservation of baryonic number 2 Breaking of symmetry between particles and 3 Deviation from thermal equilibrium List of baryogenesis scenarios: Heavy particle decays (Sakharov 1967) Baryogenesis by evaporation (Hawking 1974, Zeldovich 1976, Dolgov 1980) Electroweak baryogenesis (Kuzmin et al. 1985) SUSY condensate baryogenesis (Affleck, Dine 1985) Baryo-through- (Fukugita, Yanagita 1986) Spontaneous baryogenesis (Cohen, Kaplan 1987) Gravitational baryogenesis (Davoudiasl et al. 2004) Baryogenesis due to CPT violation (Dolgov 2010) SBG and GBG do not demand an explicit C and CP violation and can proceed in thermal equilibrium. E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 5 / 27 Spontaneous Baryogenesis (SBG)

Cosmological asymmetry can be created by SBG in thermal equilibrium: A. Cohen, D. Kaplan, Phys. Lett. B 199, 251 (1987); Nucl.Phys. B308 (1988) 913. A.Cohen, D.Kaplan, A. Nelson, Phys.Lett. B263 (1991) 86-92 Reviews: A.D.Dolgov, Phys. Repts 222 (1992) No. 6; V.A. Rubakov, M.E. Shaposhnikov, Usp. Fiz. Nauk, 166 (1996) 493; A.D. Dolgov, Surveys in High Energy Physics, 13 (1998) 83. The term ”spontaneous” is related to spontaneous breaking of underlying symmetry of the theory. Unbroken phase: the theory is invariant with respect to the global U(1)-symmetry, which ensures conservation of total baryonic number. Spontaneous symmetry breaking: the Lagrangian density acquires the term

µ LSB = (∂µθ)JB µ θ is the (pseudo) Goldstone field and JB is the baryonic current of matter fields. E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 6 / 27 µ SBG: LSB = (∂µθ)JB Spatially homogeneous field θ = θ(t):

˙ 0 LSB = θ nB , nB ≡ JB

nB is the baryonic number density, so it is tempting to identify θ˙ with the chemical potential, µB , of the corresponding system. The identification of θ˙ with chemical potential is questionable. It depends upon the representation chosen for the fermionic fields and is heavily based on the assumption θ˙ ≈ const: A.D. Dolgov, K. Freese, Phys.Rev. D 51 (1995) 2693-2702; A.D. Dolgov, K. Freese, R. Rangarajan, M. Srednicki, Phys.Rev. D 56 (1997) 6155 E.V. Arbuzova, A.D. Dolgov, V.A. Novikov, Phys. Rev. D 94 (2016) 123501 – the assumption θ˙ ≈ const is relaxed. Still the scenario is operative and presents a beautiful possibility to create an excess of particles over antiparticles in the Universe.

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 7 / 27 Gravitational Baryogenesis (GBG)

Subsequently the idea of gravitational baryogenesis was put forward: H. Davoudiasl, R. Kitano, G. D. Kribs, H. Murayama, P. J. Steinhardt, Phys. Rev. Lett. 93 (2004) 201301, hep-ph/0403019. The scenario of SBG was modified by the introduction of the coupling of the baryonic current to the derivative of the curvature scalar R:

1 µ LGBG = (∂µR)J M2 B where M is a constant parameter with the dimension of . The addition of the curvature dependent term to the Hilbert-Einstein Lagrangian of GR leads to higher order gravitational equations of motion which are strongly unstable with respect to small perturbations.

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 8 / 27 Gravitational Baryogenesis with Scalar Baryons

EA, A.D. Dolgov, ”Intrinsic problems of the gravitational baryogenesis”, Phys.Lett. B769 (2017) 171-175, arXiv:1612.06206.

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 9 / 27 GBG with scalars

The action of the scalar model has the form: Z √ m2 1  A = d4x −g Pl R + (∂ R)Jµ − g µν∂ φ ∂ φ∗ + U(φ, φ∗) − A 16π M2 µ µ ν m

19 where mPl = 1.22 · 10 GeV is the Planck mass, Am is the matter action. The baryonic number is carried by scalar field φ with potential U(φ, φ∗). If the potential U(φ) is not invariant w.r.t. the U(1)-rotation φ → eiβφ =⇒ the baryonic current defined in the usual way is not conserved. ∗ ∗ Jµ = iq(φ ∂µφ − φ∂µφ ) Here q is the baryonic number of φ The corresponding equation of motion for the curvature scalar: m2 1 1 Pl R + (R + 3D2)D Jα + Jα D R − D φ Dαφ∗ + 2U(φ) = − T µ 16π M2 α α α 2 µ

Dµ is the covariant derivative, Tµν is the energy-momentum tensor of matter E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 10 / 27 GBG with scalars: non-invariant potential U(φ)

The equation of motion for field φ is:

2 ∂U iq µ 2 D φ + = − (2DµRD φ + φD R) ∂φ∗ M2 According to definition, the current divergence is: 2   µ 2q  ∗ µ µ ∗ 2 2  ∂U ∗ ∂U DµJ = DµR (φ D φ + φD φ ) + |φ| D R + iq φ − φ M2 ∂φ ∂φ∗ If U = U(|φ|), the last term disappears, but the current is non-conserved. this non-conservation does not lead to any cosmological it can produce or annihilate an equal number of baryons and antibaryons, since the current non-conservation is bilinear in terms of φ∗φ. To create cosmological baryon asymmetry we need new types of interactions, e.g. 4 ∗ ∗4 U4 = λ4φ + λ4 φ This potential is non invariant w.r.t. the phase rotation of φ and can induce the B-non-conserving process of transition 2φ → 2φ¯. E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 11 / 27 EoM in FRW background: ds2 = dt2 − a2(t)dr2

In the homogeneous case the equation for the curvature scalar:

2 (tot) mPl 1 h 2 α 0i T R + (R + 3∂ + 9H∂t )DαJ + RJ˙ = − 16π M2 t 2 J0 is the baryonic number density of the φ-field, H = a˙/a (Hubble par.) T (tot) is the trace of the energy-momentum tensor of matter including contribution from the φ-field. In the homogeneous and isotropic cosmological

T (tot) = % − 3P ,

where % and P are the energy density and the pressure of the plasma. Relativistic plasma: 2 4 % = π g∗T /30 with T and g∗ being respectively the plasma temperature and the number of particle species in the plasma. 2 2 4 2 The Hubble parameter: H = 8π%/(3mPl ) ∼ T /mPl

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 12 / 27 Equation for R: the fourth order

The covariant divergence of the current in the homogeneous case: 2   α 2q h ∗ ∗ ∗ i ∂U ∗ ∂U DαJ = R˙ (φ φ˙ + φφ˙ ) + (R¨ + 3HR˙ ) φ φ + iq φ − φ M2 ∂φ ∂φ∗ The expectation values of the products of the quantum operators φ, φ∗, and their derivatives after the thermal averaging: T 2 hφ∗φi = , hφ∗φ˙ + φ˙ ∗φi = 0 12 Equation of motion for the classical field R in the cosmological plasma: m2 q2 h  i 1 T (tot) Pl R + R +3 ∂2 + 9H∂  R¨ + 3HR˙ T 2 + R˙ hJ0i = − 16π 6M4 t t M2 2 hJ0i is the thermal average value of the baryonic number density of φ. This term can be neglected, since it is small initially and subdominant later. NB: We obtained the 4th order differential equation for R. E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 13 / 27 Exponential Solutions

Keeping only the linear in R terms and neglecting higher powers of R, such as R2 or HR, we obtain the linear fourth order equation:

d 4R 1 m2 M4 + µ4R = − T (tot) , µ4 = Pl dt4 2 8πq2T 2 The homogeneous part of this equation has exponential solutions:

R ∼ eλt , λ = |µ|eiπ/4+iπn/2 , n = 0, 1, 2, 3

There are two solutions with positive real parts of λ. Curvature scalar is exponentially unstable w.r.t. small perturbations, so R should rise exponentially fast with time and quickly oscillate around this rising function.

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 14 / 27 The characteristic rate of the perturbation explosion is much larger than the rate of the universe expansion, if:  2 6 2 8 4 4 8π% 16π g∗ T (Re λ) > H = 2 = 4 3mPl 2025 mPl 2 4 % = π g∗T /30 is energy density of the primeval plasma at temperature T

g∗ ∼ 10 − 100 is the number of relativistic degrees of freedom in the plasma. This condition is fulfilled if 2025 m6 M4 Pl > 1 orT m3/5M2/5 9 7 2 2 10 . Pl 2 π q g∗ T Let us stress that at these the instability is quickly developed and the standard would be destroyed. To preserve the successful BBN results =⇒ impose the condition that the development of the instability was longer than the Hubble time at the BBN epoch at T ∼ 1 MeV =⇒ M should be extremely small: M < 10−32 MeV =⇒ A tiny M leads to a huge strength of coupling

1 µ LGBG = (∂µR)J M2 B It surely would lead to pronounced effects in stellar physics. E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 15 / 27 Gravitational Baryogenesis with Fermions

EA, A.D. Dolgov, ”Instability of gravitational baryogenesis with fermions”, JCAP 1706 (2017) no.06, 001, arXiv:1702.07477.

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 16 / 27 GBG with fermions

We start from the action in the form: Z " 2 # 4 p mPl A = d x −g R − Lm 16π with

i µ µ Lm = (Q¯γ ∇µQ − ∇µQ¯ γ Q) − mQ QQ¯ 2 i µ µ + (L¯γ ∇µL − ∇µL¯γ L) − mLLL¯ 2 g f  ¯ c ¯ ¯c ¯  µ + 2 (QQ )(QL) + (Q Q)(LQ) + 2 (∂µR)J + Lother mX m0 Q is the quark (or quark-like) field with non-zero baryonic number, L is another fermionic field ()

∇µ is the covariant derivative of Dirac fermion in tetrad formalism Jµ = Q¯γµQ is the quark current with γµ being the curved space gamma-matrices; Lother describes all other forms of matter.

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 17 / 27 The four-fermion interaction between quarks and g  ¯ c ¯ ¯c ¯  2 (QQ )(QL) + (Q Q)(LQ) mX is introduced to ensure the necessary non-conservation of the baryonic number g is a dimensionless coupling constant.

mX is a constant parameter with dimension of mass, which may be of the order of 1014 − 1015 GeV in grand unified theories. The gravitational EoM can be written as: 2   mPl 1 Rµν − gµν R = −gµν Lm + 8π 2 i   (Q¯(γµ∇ν + γν ∇µ)Q − (∇ν Q¯ γµ + ∇µQ¯ γν )Q + 4 i   (L¯(γµ∇ν + γν ∇µ)L − (∇ν L¯ γµ + ∇µL¯ γν )L − 4 2f  2  α f 2 Rµν + gµν D − DµDν DαJ + 2 (Jµ∂ν R + Jν ∂µR) m0 2m0 where Dµ is the usual tensor covariant derivative in background metric. E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 18 / 27 Gravitational EoM for Trace

Taking trace with respect to µ and ν we obtain:

m2 2g Pl ¯ ¯  ¯ c ¯ ¯c ¯  − R = mQ QQ + mLLL + 2 (QQ )(QL) + (Q Q)(LQ) 8π mX 2f 2 α − 2 (R +3 D )DαJ + Tother m0

Tother is the trace of the energy momentum tensor of all other fields.

At relativistic stage we can take Tother = 0. To calculate the current divergence, we use kinetic equation, which leads α to an explicit dependence of DαJ on R , if the current is not conserved. As a result we obtain high (fourth) order equation for R. We study solutions in cosmology in homogeneous and isotropic FRW background:

2 2 2 2 α t ds = dt − a (t)dr , DαJ = (∂t + 3H)J ≡ (∂t + 3H)nB

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 19 / 27 Equilibrium w.r.t. Elastic Scattering and Annihilation

Equilibrium distribution functions of quarks and leptons: 1 f = ≈ e−E/T +ξ e(E/T −ξ) + 1 ξ = µ/T is dimensionless chemical potential, different for quarks and leptons The assumption of kinetic equilibrium is well justified since it is usually enforced by very efficient elastic scattering. The baryonic number density is given by the expression: Z d 3q n = (f − f ) B 3 q q¯ 2Eq (2π) 3 ! 3 3 ! gS Bq µ gS Bq T ξ = µT 2 + = ξ + 6 π2 6 π2 T is the cosmological plasma temperature

gS and Bq are respectively the number of the spin states and the baryonic number of quarks.

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 20 / 27 Kinetic Equation: the reaction q1 + q2 ↔ q¯3 + l4

t The kinetic equation for the variation of the baryonic number density nB ≡ J coll (∂t + 3H)nB = IB

In the case of small dimensionless chemical potentials of quarks, ξq , and leptons, ξl : 2 8 " ˙ # coll CI g T 3f R(t) IB ≈ 4 2 − 3ξq + ξl mX m0 T

CI is a positive dimensionless constant

The conservation of the sum of baryonic and leptonic numbers gives: ξl = −ξq /3 NB. Here we consider only the simple situation with quasi-stationary background: R(t) ≈ R˙ (t) t The case of an essential variation of R˙ (t) is analogous to fast variation of θ˙(t) studied in EA, A.D. Dolgov, V.A. Novikov, Phys. Rev. D 94 (2016) 123501 “General properties and kinetics of spontaneous baryogenesis” It is much more complicated technically and is postponed for the future work. E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 21 / 27 Stationary Point Approximation

For small chemical potential the baryonic number density is equal to

gs Bq 3 nB ≈ ξq T 6 If the temperature adiabatically decreases in the course of the cosmological expansion, according to T˙ = −HT , the kinetic equation coll (∂t + 3H)nB = IB turns into " # 9f R˙ (t) ˙ 2 5 4 ξq = Γ 2 − ξq , Γ ∼ g T /mX 10m0 T where Γ is the rate of B-nonconserving reactions. If Γ is large, this equation can be solved in stationary point approximation as 9 f R˙ eq ˙eq eq ξq = ξq − ξq /Γ , where ξq = 2 . 10 m0 T eq If we substitute ξq into EoM for trace we arrive to the 4th order equation for R. E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 22 / 27 Curvature Instability

The contribution of thermal matter into EoM for trace can be neglected 2 mPl 2f 2 R = 2 (R +3 D )(∂t + 3H)nB 8π m0 From the kinetic equation ˙ eq 9 f R nB ∼ ξq = 2 10 m0 T Neglecting the H-factor in comparison with time derivatives of R, we arrive to: d 4R = λ4R (∗) dt4 where 4 2 4 2 2 λ = CλmPl m0 /T with Cλ = 5/(36πf gs Bq ). Equation (∗) has extremely unstable solution with instability time by far shorter than the cosmological time. This instability would lead to an explosive rise of R, which may possibly be terminated by the nonlinear terms ∼to the product of H to lower derivatives of R. E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 23 / 27 Possible Stabilization

Correspondingly one may expect stabilization when HR ∼ R˙ , i.e. H ∼ λ Since H˙ + 2H2 = −R/6 H would also exponentially rise together with R, H ∼ exp(λt) and λH ∼ R Thus stabilization may take place at 2 2 R ∼ λ ∼ mPl m0/T which is much larger than the normal General Relativity value 2 RGR ∼ Tmatter /mPl

where Tmatter is the trace of the energy-momentum tensor of matter. E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 24 / 27 Conclusions

The derivative coupling of baryonic current to the curvature scalar in GBG scenarios leads to higher (4th) order equations for gravitational field. These equations are unstable with respect to small perturbations of the FRW background and such instability leads to an exponential rise of the curvature. For a large range of cosmological temperatures the development of the instability is much faster than the Universe expansion rate. The rise of R could be terminated by the non-linear terms in the EoM, if the non-linear terms become comparable by magnitude with the linear ones. It means that the rise terminates when R by far exceeds the value found in the normal General Relativity. µ Simple versions of GBG based on the coupling, ∼ (∂µR)JB , would be incompatible with observations and a stabilization mechanism is desirable. Probably stabilization may be achieved in a version of F (R)-gravity or by an introduction of a form-factor g(R) into the coupling LGBG , such that g(R) drops down with rising R.

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 25 / 27 The END

Thank You for Your Attention

E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 26 / 27 Kinetic Equation: the reaction q1 + q2 ↔ q¯3 + l4

t The kinetic equation for the variation of the baryonic number density nB ≡ J through this reaction in the FRW background has the form: coll (∂t + 3H)nB = IB where the collision integral for space and time independent interaction is equal to: Z coll 4 4 IB = −3Bq (2π) dνq1,q2 dνq¯3,l4 δ (q1 + q2 − q3 − l4)

 2 2  |A(q1 + q2 → q¯3 + l4)| fq1 fq2 −| A(¯q3 + l4 → q1 + q2)| fq¯3 fl4 A(a → b) is the amplitude of the transition from state a to state b

Bq is the baryonic number of quark, fa is the phase space distribution (the occupation number), and the element of phase space is:

3 3 d q1 d q2 dν = q1,q2 3 3 2Eq1 (2π) 2Eq2 (2π)

p 2 2 where Eq = q + m is the energy of particle with three-momentum q and mass m. E. Arbuzova Impact of GBG on cosmological evolution 1 June 2018 27 / 27