Interaction of the Axionic Dark Matter, Dynamic Aether, Spinor and Gravity ﬁelds As an Origin of Oscillations of the Fermion Effective Mass

Eur. Phys. J. C (2021) 81:674 https://doi.org/10.1140/epjc/s10052-021-09341-z

Interaction of the axionic dark matter, dynamic aether, spinor and gravity ﬁelds as an origin of oscillations of the fermion effective mass

Alexander B. Balakina, Anna O. Efremovab Department of General Relativity and Gravitation, Institute of Physics, Kazan Federal University, Kremlevskaya Str. 16a, Kazan 420008, Russia

Received: 27 April 2021 / Accepted: 12 June 2021 / Published online: 30 July 2021 © The Author(s) 2021

Abstract In the framework of the Einstein–Dirac-axion- with a half-integer spin, which do not coincide with their aether theory we consider the quartet of self-interacting cos- antiparticles. This subclass of fermions contains, in particu- mic fields, which includes the dynamic aether, presented lar, the baryons (e.g., protons and neutrons), the leptons (e.g., by the unit timelike vector field, the axionic dark mater, electrons, positrons, massive neutrinos). It is especially inter- described by the pseudoscalar field, the spinor field asso- esting, when one deals with the supernova and solar neutri- ciated with fermion particles, and the gravity field. The nos, with neutrino oscillations and the problem of neutrino key, associated with the mechanism of self-interaction, is masses (see, e.g., [1,2] for details). When one deals with installed into the modified periodic potential of the pseu- the massless fermions, the Weyl and Majorana equations are doscalar (axion) field constructed on the base of a guid- used, as specific particular cases of the Dirac equations. ing function, which depends on one invariant, one pseudo- The second element of the theory is the axionic dark mat- invariant and two cross-invariants containing the spinor and ter, the key participant of the cosmological events (see, e.g., vector fields. The total system of the field equations related [3–5] for references, historical motives and mathematical to the isotropic homogeneous cosmological model is solved; details). We follow the idea of the author of the review [3], we have found the exact solutions for the guiding function for that now just the axions are considered to be the leading three cases: nonzero, vanishing and critical values of the cos- particle candidates to provide the phenomena attributed to mological constant. Based on these solutions, we obtained the the cosmic dark matter (e.g., the flat galactic rotation curves, expressions for the effective mass of spinor particles, inter- the structure and properties of the dark matter halos, etc.). acting with the axionic dark matter and dynamic aether. This New aspects of this discussion and new argumentation can effective mass is shown to bear imprints of the cosmological be found in the recent review [6]. epoch and of the state of the cosmic dark fluid in that epoch. The dynamic aether, the third key element of the theory, was introduced in [7–9]; it can produce the effects, which are usually associated with the influence of the cosmic dark 1 Introduction energy [10–12]. Unexpectedly interesting applications appear, when one The Einstein–Dirac-axion-aether theory is very effective considers the pairwise interactions, e.g., between spinors and instrument for bringing together various trends in modern dark matter, spinors and dark energy, dark matter and aether, Cosmology, Astrophysics and High Energy Physics. First of etc. For instance, the enigma of the neutrino masses pushed all, this theory deals with the Dirac spinor fields, and thus to the study of coupling between neutrinos and dark matter, it can be applied to description of wide class of phenom- provoking an extensive series of works (see, e.g., [13–20]). ena, which occur in the fermion systems. The Dirac spinor The theory of coupling between spinor field and dark energy fields, according to the classification adopted in the Standard gives another example of such pairwise interaction (see, e.g., model, describe the Dirac fermions, i.e., massive particles [21–26]). In this sense, it is interesting to mention the new trend appeared in the theory of interactions in the Dark Sec- tor: we mean the so-called dark spinors (see, e.g., [27–29]). a e-mail: [email protected] (corresponding author) Extensions of the canonic Einstein-aether theory [8]have b e-mail: [email protected] 123 674 Page 2 of 12 Eur. Phys. J. C (2021) 81 :674 given many interesting results for the models of coupling of the axion field V (φ, ∗(S, P,...)), which we use for the between aether and scalar field (see, e.g., [30,31]), for the description of this model. models of interaction of the aether with the axion field [32– Our final remark concerns the electromagnetic field and 34], and for the aether models including the electromagnetic Yang-Mills fields. We assume here that these gauge U(1) field [35,36]. and SU(N) symmetric fields are vanishing, and thus, the There are examples of investigations of the triple inter- extended derivative of the Dirac spinor field does not contain actions: for instance, in the work [37] the coupling between the corresponding potentials. The problem is that there exists spinor, axion and scalar fields is studied; in the papers [38,39] the aether-like representation of the gauge field [50,51], and the coupling between aether, axions and photons is consid- the question arises: whether the aether velocity four-vector k k ered; in the works [40–42] the spinor and scalar fields are con- U or its SU(N) generalizations U(a) (see, e.g., [52]) can play sidered to be coupled to the fluids (perfect, viscous, magne- the role of the gauge potential? We assume that this question tized). In some works the spinor field is presented in terms of is still open, but in this work we follow the classic non-gauge the nonlinear and nonminimal formalisms (see, e.g., [43,44]). invariant version of the dynamic aether. In the context of these numerous investigations, we have The construction of the axion field potential V (φ, ∗) to clarify: what are the aims and frames of our work. First admits the following twofold interpretation: on the one hand, of all, we have to say that many quoted results are for- the spinor field and the aether regulate the behavior of the mulated on the language of High Energy Physics, but we axion field via this potential; on the other hand, the axionic use the formalism of the Field Theory, to be more precise, backreaction modifies the Dirac equations for the spinor field, we use the formalism of the Einstein–Dirac-axion-aether and the effective spinor mass matrix M appears instead of theory. In the framework of this theory we consider the the intrinsic mass m. In other words, we assume that if the quadruple coupling of the gravitational, spinor, pseudoscalar intrinsic mass of the spinor particle (e.g., of the neutrino) and vector fields. In the total Lagrangian of the system the is equal to zero, m = 0, then the coupling to the axionic gravity field is presented by the classical Einstein-Hilbert dark matter produces an effective mass, which depends, e.g., term; in the nearest future we plan to use the frames of the on cosmological epoch and on the state of the axionic dark Modified theories of gravity along the line proposed in the matter in that epoch. works [45–48]. The spinor field ψ (massive or massless) The paper is organized as follows. In Sect. 2.1 we describe is described by the extended Dirac equations. The pseu- the action functional of the model; in Sect. 2.2 we specify doscalar field φ is associated with the axionic dark mat- the structure and properties of the modified periodic poten- ter. The unit timelike vector field U i describes the veloc- tial of the axion field; in Sect. 2.3 based on the Lagrange ity of the dynamic aether flow. We assume that the potential formalism we derive the extended equations for the vector, of the axion field V (φ, ∗) contains the guiding function axion, spinor and gravitational fields. Section 3 contains cos- ∗, which depends on four arguments (S, P,ω,).The mological application of the elaborated model. In Sect. 3.1 first argument S = ψψ¯ is the invariant constructed using the we reduce the field equations of the model to the symme- ¯ 5 Dirac spinor; the pseudoinvariant P = ψγ ψ contains the try associated with the isotropic homogeneous cosmology. 5 ¯ k Dirac matrix γ ; the scalar ω = Uk ψγ ψ is the cross- In Sect. 3.2 we obtain exact solutions of the complete set invariant, which contains both spinor and vector fields; the of the reduced field equations, and discuss the properties of ¯ 5 k pseudoscalar = Uk ψγ γ ψ also has to be indicated as the exact solutions to the guiding function ∗ for two cases: the cross -pseudoinvariant. with and without cosmological constant, = 0 and = 0. In this work, classifying the scalars, which could be the In Sect. 4 we discuss the properties of the obtained effective arguments of the guiding function, we omitted the scalars, mass attributed to the spinor field coupled to the axionic dark obtained from the decomposition of the covariant deriva- matter. Section 5 contains discussion and conclusions. i tive of the aether velocity four-vector ∇kU : the expansion k scalar =∇kU , the square of the acceleration four-vector j k j a = U ∇kU , the squares of the symmetric shear tensor σ jk and of the skew-symmetric vorticity tensor ω jk (see [32,34] for details). We hope to extend the model, respec- 2 The formalism of the Einstein–Dirac-aether-axion tively, in the next work, but now we restrict our-selves by the theory scalars of zero order in derivative, according to the terminol- ogy of the Effective field theory [49]. Keeping in mind this 2.1 Action functional idea, we do not include into the Lagrangian the pseudoscalar ¯ k ¯ k 5 ∇kφ(ψγ ψ), and the scalar ∇kφ(ψγ γ ψ). As for the zero The action functional describing the system of interacting order cross-scalar P ·φ and cross-pseudoscalar S ·φ,itseems gravitational, spinor, pseudoscalar (axion) vector fields can to be not logical to include them into the specific potential be written as follows: 123 Eur. Phys. J. C (2021) 81 :674 Page 3 of 12 674 √ 4 1 m n heavy spinor particles, and thus they also can be described − S = d x −g R + 2 + λ gmnU U − 1 2κ by the Dirac equations, if we work in the paradigm of the + ab ∇ m∇ n + 1 2 −∇kφ∇ φ Field Theory. Clearly, in general case in order to describe a K mn aU bU 0 V k 2 multi-fermion system, we have to introduce the correspond- i ¯ k ¯ k ¯ ing multi-component spinors, instead of the four-component + [ψγ Dkψ − Dkψγ ψ]−mψψ − L( ) . 2 baryon one. There exists an alternative to this approach: one can (1) consider the baryonic matter in terms of a multi-component fluid with the stress-energy tensor of the perfect fluid. It is κ As usual, g is the determinant of the metric, is the Einstein necessary to use this approach, e.g., when the spinor field, constant, R is the Ricci scalar, is the cosmological constant, which we analyze, presents the massless neutrino. ∇k is the covariant derivative. λ is the Lagrange multiplier, U i is the unit timelike vector field, which is associated with 2.2 The structure of the potential of the axion field the velocity four-vector of the aether flow. The object ab = ab + δa δb + δaδb + a b We assume that the modified periodic potential of the axion K mn C1g gmn C2 m n C3 n m C4U U gmn (2) field is given by the function contains four phenomenological constants C1, C2, C3, C4 2 2 πφ and presents the so-called constitutive tensor in the model of m A ∗ 2 V (φ, ∗) = 1 − cos . (8) the dynamic aether [8]. 2π 2 ∗ The dimensionless pseudoscalar field φ is associated with This potential is indicated as periodic, since V (φ + n∗) = the axions, hypothetical massive pseudo-Goldstone bosons V (φ); it has the minima at φ=n∗. Near the minima, when [53–55]; V is the potential of the pseudoscalar field, and the φ → + φ˜ |φ˜| n ∗ , and is small, the potential takes the stan- parameter 0 is connected with the constant of the axion- → 2 φ˜2 1 dard form V m A , where m A is the axion mass. The photon coupling gAγγ by the relationship = gAγγ. 0 function ∗ plays the role of vacuum average value of the The matrix-column ψ describes the spinor field, ψ¯ is its axion field; we assume that it is not a constant, and it depends Dirac conjugated field, ψ¯ = ψ∗T γ 0. The matrices γ k satisfy on coordinates via some auxiliary invariants. In the works the relationships [32,33] these invariants were constructed using the covari- γ mγ n + γ nγ m = 2Egmn, (3) ant derivative of the velocity four-vector, associated with the dynamic aether flow. In the papers [57,58]wehaveusedthe × γ k where E is the unit 4 4 matrix. The matrices are con- moduli of the Killing vectors as auxiliary invariants, the argu- γ (a) nected with the standard constant Dirac matrices via the ments of the guiding function ∗. Now we develop this idea m γ k = k γ (a) tetrad four-vectors X(a) by the convolution X(a) and assume that the guiding function ∗ can depend on the ( ) with respect to the tetrad index a . The constant Dirac matri- spinor field via the following four invariants (two scalars and ces satisfy the relationships two pseudoscalars): (a) (b) (b) (a) (a)(b) γ γ + γ γ = 2Eη , (4) = ψψ,¯ = ψγ¯ 5ψ, S P ( )( ) η a b ¯ k ¯ 5 k where is the Minkowski metric. The quartet of the ω = Uk ψγ ψ ,= Uk ψγ γ ψ . (9) m tetrad four-vectors X(a) is known to satisfy two normalization– orthogonality conditions: We use the following definition for the matrix γ 5. m n (a)(b) m n mn = η( )( ),η = . gmn X(a) X(b) a b X(a) X(b) g (5) 5 1 m n p q γ = mnpqγ γ γ γ , (10) 4! The term Dk defines the spinor covariant derivatives, which are given by where mnpq is the Levi-Civita tensor expressed via the Levi- Civita antisymmetric symbol E as follows: ψ = ∂ ψ − ψ, ψ¯ = ∂ ψ¯ + ψ¯ . mnpq Dk k k Dk k k (6) √ mnpq = −gEmnpq, E0123 =−1. (11) The Fock–Ivanenko matrices k [56] can be expressed via the covariant derivatives of the tetrad four-vectors: The matrix γ 5 does not depend on the metric, since √ 1 (a) s n m 5 0 1 2 3 (0) (1) (2) (3) (5) k = gmn X γ γ ∇k X( ). (7) γ = −gγ γ γ γ = γ γ γ γ = γ . (12) 4 s a The parameter m describes the mass of the spinor field. Two last quantities in (9), ω and , can be indicated as cross- Finally, we would like to focus on the term L(baryon),the invariant and cross - pseudoinvariant, respectively, since they last element of the action functional (1). Formally speaking, contain quantities, which characterize both the aether flow the baryons (protons, neutrons, etc.) are the fermions, the and spinor field. Let us mention that in many aspects we 123 674 Page 4 of 12 Eur. Phys. J. C (2021) 81 :674 follow the approach presented in the series of works [40– 2.3.2 Equations for the axion field 44]; the authors of these works have focused on the coupling k of the scalar and spinor fields via the kinetic term ∇ ϕ∇kϕ · Variation of the action functional (1) with respect to the axion F(S, P,...), so one can say that we extend this approach field gives the equation considering the axion field instead of scalar, introducing the 2 πφ vector field related to the dynamic aether, and modifying the k m A ∗ 2 ∇ ∇kφ =− sin . (19) potential V instead of the kinetic term. 2π ∗

The spinor field predetermines the structure of the function 2.3 Model field equations ∗(S, P, H,), and thus it regulates the behavior of the axion field. 2.3.1 Equations for the aether field 2.3.3 Equations for the spinor field Variation of the action functional (1) with respect to the λ i Lagrange multiplier and to the four-vector U gives, respec- Variation of the action functional (1) with respect to the quan- tively, the normalization condition tities ψ¯ and ψ gives, respectively,

m n = , n ¯ n ¯ gmnU U 1 (13) iγ Dnψ − Mψ = 0, iDnψγ + ψ M = 0, (20) and the dynamic equation for the aether ﬂow where the matrix M is presented by the formula

= − ∇ J aj = λU j + C DU ∇ j U m + κ I j . (14) M mE T a 4 m ∂ ∂ ∂ ∂ ∗ ∗ 5 ∗ ∗ 5 k × E + γ + E + γ Ukγ . J a ∂ S ∂ P ∂ω ∂ The tensor j is standardly presented as (21) J a = K ab ∇ U n j jn b If we consider the effective mass as the scalar function, we = ∇a + δa + ∇ a + a , C1 U j C2 j C3 j U C4U DU j (15) can use the formula ¯ k (ψ Mψ) with the convective derivative D = U ∇k. The four-vector M≡ (ψψ)¯ I j is given by T ∂∗ ∂∗ ∂∗ ∂∗ = m − S + P + ω + . (22) m2 2∗ 2πφ πφ 2πφ S ∂ S ∂ P ∂ω ∂ I j = A 0 1 − cos − sin 2π 2 ∗ ∗ ∗ According to (20), (21) the axion ﬁeld φ and the aether veloc- ∂ ∂ × ∗ ψγ¯ j ψ + ∗ ψγ¯ 5γ j ψ . ity form an effective mass of the spinor ﬁeld. ∂ω ∂ (16)

Below, keeping in mind the compactness of formulas, we use 2.3.4 Equations for the gravity ﬁeld many times the function T = T (φ, ∗), deﬁned as follows: Variation of the action functional (1) with respect to the met-

2 2 ric yields m ∗ 2πφ πφ 2πφ T = A 0 1 − cos − sin . 2π 2 ∗ ∗ ∗ 1 U (A) (D) (B) Rpq − gpq R − gpq = T + κT + κT + κT . (17) 2 pq pq pq pq (23) j Thus the four-vector I can be shortly rewritten as ( ) The standard stress-energy tensor of the aether T U is known pq ∂∗ ∂∗ to have the form: I j = T ψγ¯ j ψ + ψγ¯ 5γ j ψ . ∂ω ∂ (18) (U) = 1 abmn∇ ∇ + λ Tpq gpq K aUm bUn UpUq = (φ, ) 2 The function T T ∗ takes zero value, when the m +∇ U(pJq)m − Jm(pUq) − J(pq)Um axions are in the equilibrium state, φ=n∗, and the inte- + (∇ )(∇m ) − (∇ )(∇ m) ger n, describing the serial number of the equilibrium level, C1 mUp Uq pUm qU is arbitrary. +C4 DUp DUq . (24) 123 Eur. Phys. J. C (2021) 81 :674 Page 5 of 12 674

The stress-energy tensor of the axion ﬁeld contains one stan- fermion masses. We see that this problem can be success- dard and one new elements: fully solved also in the framework of the anisotropic Bianchi

models, however we focus now on the Friedmann-type model (A) 2 1 k T = ∇pφ∇q φ + gpq V −∇kφ∇ φ with the cosmological constant in order to clarify physical pq 0 2 aspects of the problem. 1 ∂∗ + T U ψγ¯ ψ + U ψγ¯ ψ ∂ω p q q p 2 3.1 Reduced field equations ∂ ∗ ¯ 5 ¯ 5 + Up ψγ γq ψ +Uq ψγ γpψ . (25) ∂ 3.1.1 Metric, tetrad and connection coefficients The new element in (25), which is proportional to the function T , is originated from the variation of the potential V with For description of the isotropic homogeneous spacetime we respect to the metric. Here we used the formulas use the metric 2 = 2 − 2( )( 2 + 2 + 2), δX n ds dt a t dx dy dz (32) (a) 1 n n = X ( )δ + X ( )δ , (26) ˙ δ pq p a q q a p ( ) = a g 4 the Hubble function H t a , and the set of the tetrad δg 1 four-vectors mn =− g g + g g , (27) δ pq mp nq np mq g 2 i = i = δi , i = δi 1 , X(0) U 0 X(1) 1 δγ (a) δγ 5 δU j a(t) = 0, = 0, = 0. (28) δ pq δ pq δ pq 1 1 g g g Xi = δi , Xi = δi . (33) (2) 2 a(t) (3) 3 a(t) Details of variation procedure for the tetrad four-vectors can be found, e.g., in [59,60]. The spinor connection coefficients k have now very simple (D) form The stress-energy tensor of the spinor field Tpq is 1 ( ) ( ) = 0,= a˙γ 1 γ 0 , (D) i ¯ 0 1 T =−gpq L( ) + ψγp Dq ψ 2 pq S 4 1 (2) (0) 1 (3) (0) +ψγ¯ ψ − ( ψ)γ¯ ψ − ( ψ)γ¯ ψ . 2 = a˙γ γ ,3 = a˙γ γ . (34) q Dp Dp q Dp q (29) 2 2 As a direct consequence of (34) we obtain the auxiliary for- The term L(S) appeared in (29) mula i ¯ k ¯ k ¯ 3 L( ) = ψγ Dkψ − (Dkψ)γ ψ − mψψ , (30) k 0 k S γ k =− Hγ =−kγ . (35) 2 2 takes the following form on the solutions to the modified ¯ 3.1.2 Reduced evolutionary equation for the unit vector Dirac equations: L( ) = ψ(M − m)ψ. Now, in contrast to S field the classical case, when ∗ is constant, the effective mass matrix M (see (21)) does not coincide with the intrinsic mass The symmetry of the model requires that the aether velocity matrix mE, and the term L( ) does not vanish. S four-vector has to have the form U j = δ j , and thus the The last term in the right-hand side of (23) describes the 0 covariant derivative is contribution of the baryonic matter, if it is considered as the ∇ m = ( ) δmδ1 + δmδ2 + δmδ3 . perfect fluid: kU H t 1 k 2 k 3 k (36) ( ) B = ( + ) − , k Tpq W Vp Vq gpq (31) The corresponding acceleration four-vector vanishes DU = 0, and the expansion scalar =∇ U k is equal to = 3H. where W and are the baryonic energy density and pres- k Thus, the tensor J aj is now symmetric, and its components sure, respectively, and the velocity four-vector of the fluid, p (B) can be written as follows: V , is the unit timelike eigen-vector of the tensor Tpq , i.e., (B) p a = ( + ) δaδ1 + δaδ2 + δaδ3 + δa . Tpq V = WVq (the Landau–Lifshitz definition). J j H C1 C3 1 j 2 j 3 j 3C2 j (37) The equations for the unit vector field (14) take the form 3 Application: the model of isotropic homogeneous Universe δ0 ˙ − ( + ) 2 = λδ0 + κ . 3 j C2 H C1 C3 H j I j (38) The isotropic homogeneous cosmological model is a good Clearly, due to the model isotropy, the spatial components starting point for analysis of the problem of the induced of the four-vector I j have to be equal to zero, Iα=0, where 123 674 Page 6 of 12 Eur. Phys. J. C (2021) 81 :674

α=1, 2, 3. It is possible in one of the following three cases. 3.1.5 Evolution of the spinor invariants In the ﬁrst case we can require α α Keeping in mind (46), we can calculate the rate of evolution of ψγ¯ ψ = 0, ψγ¯ 5γ ψ = 0, (39) the invariants S, P, ω and . Let us demonstrate the method and obtain six conditions for the four complex spinor compo- of derivation on the example of the scalar S.First,wesee nents. The second case corresponds to the equilibrium state that φ= of the axionic dark matter, and the condition n ∗ leads d ψψ¯ = d ¯ −3 = a to I j 0 because of the vanishing of the function T (17). dt dt ∂∗ = ¯ In the third case we deal with the requirements ∂ω 0 and =−3H ψψ ∂∗ = ∗ ∂ 0, i.e., the function depends on S and P only. − d ( )2 ( )2 d = + a 3 ¯ γ 0 + γ¯ 0 When Iα 0, we obtain that only one equation for the dt dt velocity four-vector from the set (38) is nontrivial =−3H ψψ¯ + iψ¯ Mγ 0 − γ 0 M ψ. (47) ˙ 2 3 C2 H − (C1 + C3)H = λ + κ I0. (40) Using the formula for the effective mass matrix M (21) with This equation gives the Lagrange multiplier T given by (17) we can rewrite the evolutionary equation for the scalar S as follows

∂ ∂ ∂ ∂ ˙ 2 ∗ ∗ ˙ ∗ ∗ λ(t)=3 C2 H − (C1 + C3)H − κT ω + . S + 3HS =−2iT + P . (48) ∂ω ∂ ∂ P ∂ (41) Similarly, for the pseudoinvariant P we can write the evolu- This quantity contributes to the gravity ﬁeld equation via tionary equation ( ) T U (see (24)). ik P˙ + 3HP = iψ¯ Mγ 0γ 5 − γ 5γ 0 M ψ, (49)

3.1.3 Reduced evolutionary equation for the axion ﬁeld from which we obtain

∂∗ ∂∗ P˙ + 3HP =−2im + 2iT + S . (50) Field equation (19) takes now the form ∂ S ∂

m2 ∗ 2πφ ω φ¨ + 3Hφ˙ =− A sin . (42) For the scalar and pseudoscalar the results are 2π ∗ ω˙ + ω = , 3H 0 (51) Generally, this nonlinear equation with coefﬁcients depend- ˙ + =−ψ¯ γ 5 + γ 5 ψ, ing on time can be analyzed only numerically, but there are 3H i M M (52) special cases, which we discuss below. or equivalently

∂∗ ∂∗ 3.1.4 Reduced evolutionary equation for the spinor ﬁeld ˙ + 3H =−2imP + 2iT P − S . (53) ∂ S ∂ P The symmetry of the problem hints, that one can search for When the axionic dark matter is in the equilibrium state, the components of the spinor ﬁeld as the functions of time φ = n∗, and consequently, T (φ, ∗) = 0, the evolutionary only; then the Dirac equations simplify essentially equations (48)–(53) convert into

0 3 S˙ + 3HS = 0, ω˙ + 3Hω = 0, (54) iγ ∂0 + H ψ = Mψ, (43) 2 P˙ + 3HP =−2im, ˙ + 3H =−2imP. (55) 3 i ∂ + H ψγ¯ 0 =−ψ¯ M. (44) 0 2 Then we obtain the following exact solutions to these equations The replacement 3 a(t0) − 3 ¯ − 3 ¯ S(t) = S(t0) , (56) ψ = a 2 , ψ = a 2 (45) a(t)

a(t ) 3 reduces the Dirac equations, yielding ω(t) = ω(t ) 0 , (57) 0 a(t) (0) ˙ ¯˙ (0) ¯ iγ = M , i γ =− M, (46) 3 a (t0) P(t) = [P(t0) cos 2m(t − t0) where the dot denotes the derivative with respect to time. Let a3(t) us consider the consequences of these equations. +(t0) sin 2m(t − t0)] , (58) 123 Eur. Phys. J. C (2021) 81 :674 Page 7 of 12 674

3 a (t0) Below we use the auxiliary variable x defined as follows: (t) = [(t ) cos 2m(t − t ) a3(t) 0 0 a(t) d d −P(t0) sin 2m(t − t0)] . (59) x = , = xH(x) . (65) a(t ) dt dx a6(t ) 0 P2(t) + 2(t) = P2(t ) + 2(t ) 0 . (60) 0 0 a6(t) In terms of this new variable the Hubble function extracted from (61)atφ=∗ (we choose the level n = 1 as the basic Below we use this finding for reconstruction of the guiding one) is given by function ∗. 2 ˙ 2 κ[mS(t ) + W(t )] κ ∗(t0) 3.1.6 Key equation for the gravity field H(x) = + 0 0 + 0 . (66) 3 3x3 6x6

We work now in the approximation, for which the cosmolog- In terms of x the key equation for the function ∗(x) is ical baryonic matter is pressureless, = 0, the fluid velocity i = i d ˙ −4 coincides with the aether velocity, U V , and the stress- H(x) ∗(x) = ∗(t0)x , (67) p (B) dx energy tensor (31) is divergence-free, i.e., ∇ Tpq = 0. These requirements give us the known relationship for the and we are ready for integration of this key equation. ( ) 3 ( ) = ( ) a t0 cosmic dust W t W t0 a(t) . Taking into account the structure of the stress-energy tensors of the unit vector, axion 3.2.2 Solutions for the model with nonvanishing and spinor fields given by (24), (25), (29), respectively, we cosmological constant, = 0 find that the key equation for the gravity field reads When = 0, we can rewrite the Hubble function (66)as a(t ) 3 3H 2 − − κ [mS(t ) + W(t )] 0 follows: 0 0 a(t) H∞ ( ) = 6 + α 3 + β, m2 2 πφ H x x x (68) = 1κ 2 A ∗ − 2 + φ˙2 , x3 0 1 cos (61) 2 2π 2 ∗ where we introduced the following auxiliary parameters: where we introduced a new auxiliary parameter: κ[mS(t0) + W(t0)] H∞ = ,α= , 1 3 = 1 + (C1 + 3C2 + C3). (62) 2 ˙ 2 2 κ ∗(t0) β = 0 . (69) Other Einstein’s equations are the differential consequences 2 of the evolutionary equations for the aether, axion and spinor Formal integration of (67)gives fields. ∗(x) 3H∞ + const ˙ ( ) 3.2 Equilibrium axionic dark matter: how do we search for ∗ t0 the guiding function ∗? 1 x3 = √ log . (70) β 2 β(x6 + αx3 + β) + αx3 + 2β 3.2.1 Key equation for the guiding function ∗

Since x=1 when t=t0, the constant of integration can be We indicate the state of the axionic dark matter as an Equi- easily found, yielding librium state, when the potential V (φ, ∗) and its derivative ∂V ( ) = ( ) ∂φ take zero values. For the periodic potential (8) these con- ∗ x ∗ t0 ⎧ ⎫ √ ditions are satisfied, if the pseudoscalar field is in one of the ˙ ⎨ 3 ⎬ ∗(t0) x 2 β(1 + α + β) + α + 2β minima of the potential, i.e., φ = n∗, where n is an inte- + √ log . β ⎩ ⎭ ger. For the Equilibrium state the Eq. (42) converts into the 3H∞ 2 β(x6 + αx3 + β) + αx3 + 2β ( ) equation for the guiding function ∗ t : (71) ¨ ˙ ∗ + 3H∗ = 0. (63) If we are interested to find ∗ as a function of the cosmolog- This equation does not depend on the integer n and admits ical time t, we take the formula x˙ = xH(x) and immediately the first integral obtain ( ) a t 2 ( ) 3 a(t ) x dx ˙ ˙ a t0 ( − ) = 0 . ∗(t) = ∗(t0) . (64) H∞ t t0 (72) a(t) 1 x6 + αx3 + β 123 674 Page 8 of 12 Eur. Phys. J. C (2021) 81 :674

Direct integration in (72)gives present, we obtain the late-time accelerated expansion, as it

1 should be. a(t ) 3 2 ( ) = 0 β − α2 ( − ) − α . The case, when 4β=α has to be considered especially, a t 1 4 sinh [3H∞ t t∗ ] (73) 2 3 since t∗ in (75) is now infinite. We assume that 4β>α2 providing a(t) to be the real func- β= 1 α2 = tion of time; this is possible, when the positive cosmological 3.2.3 Special case 4 , 0 constant is bigger than some critical value > C , where This special case can be realized if the cosmological constant κ[mS(t ) + W(t )]2 = = 0 0 . (74) takes the critical value, C . For this special case the C 2˙ 2( ) 2 0 ∗ t0 Hubble function (68) transforms into α Also, we introduce the auxiliary time moment t∗ as H(x) = H∞ 1 + , (82) 2x3 1 2 + α t∗ = t0 − arsh . (75) and the scale factor has the form 3H∞ β − α2 4 1 α ( − ) α 3 a(t) = a(t ) 1 + e3H∞ t t0 − . (83) Asymptotic behavior of this field configuration relates to the 0 2 2 de Sitter law As for the guiding function, it is now of the form ( →∞) ∝ H∞t . a t e (76) 3 α ˙ ∗( ) x 1 + 2 t0 2 ∗(x) = ∗(t0) + log α . (84) The function ∗ tends asymptotically to the constant ∞α 3 + 3H x 2 ˙ ∗(t0) ∗(x →∞) = ∗(t ) + √ ∝ H∞t 0 β Asymptotic regime relates to the de Sitter law a e , 3H∞ √ and the guiding function tends asymptotically to the constant β( + α + β) + α + β 2˙ ∗(t ) α 2 1 2 ∗(∞) = ∗(t ) + 0 log 1 + . × log √ . (77) 0 3H∞α 2 2 β + α In this model the acceleration parameter can be recon- structed as follows Finally, it is interesting to calculate the acceleration parameter defined as x3 − α − q = α . (85) a¨ x dH x3 + − q ≡ = + . 2 2 1 (78) aH H dx The transition point is characterized by the reduced scale For the Hubble function (68) it has the form 1 factor xT = α 3 , and the corresponding time moment is 6 1 3 x − αx − 2β 1 3α − q ≡ 2 . = + . 6 3 (79) tT t0 log (86) x + αx + β 3H∞ 2 + α x Clearly, at the value of the reduced scale factor T there exists When α>1, we obtain that tT > t0. Again, asymptotically ( ) = the cosmological transition point, for which q xT 0 and the acceleration parameter tends to one, and we deal with the thus the acceleration parameter changes the sign. This point late-time acceleration, typical for the models with quasi-de is calculated to be the following: Sitter asymptotes. ⎛ ⎞ 1 One can mention that, in both cases with = 0 studied 3 α α2 above the period of cosmological time t < t < t is charac- x = ⎝ + + 2β⎠ . (80) 0 T T 4 16 terized by the decelerated expansion, i.e., after the moment t0 the inflationary stage can not be realized. This fact can Using the exact solution (73), one can rewrite this equality be explained as follows. We consider the model in which at in terms of the cosmological time t > t0 the spinor field is nonvanishing, i.e., massive fermions ⎡ ⎤ β already filled the Universe. In other words, the moment of 3 + 1 + 32 1 ⎣ α2 ⎦ the phase transition, at which the fermions were born (we tT = t∗ + arsh . (81) ∞ 4β t t < t 3H 2 − 1 indicate it as F) satisfies the inequality F 0. This means, α2 that the known inflection point on the graph of the scale fac- The function (79) shows that in the model under consider- tor evolution, which divides the epochs of the inflation and ation (we assume here that >0 and thus α>0 and the first decelerated expansion, also belongs to the interval β>0) the Universe expands with deceleration in the time t < t0. The model of such phase transition initiated by the interval t0 < t < tT.Att > tT the expansion of the Uni- dynamic aether is in progress, and we plan to extend the verse becomes accelerated, and asymptotically −q → 1. At model correspondingly in the nearest future. 123 Eur. Phys. J. C (2021) 81 :674 Page 9 of 12 674

3.2.4 Solutions for the model with vanishing cosmological tends to zero asymptotically, when x →∞. The correspond- constant, = 0, and m = 0 ing scale factor

1 ( ) = ( ) + γ( − ) 3 When = 0 and the spinor field is massive, m = 0, the a t a t0 1 3 t t0 (95) Hubble function is of the power-law type, and the guiding function ∗ H 3 ˙ H(x) = x + σ (87) ∗(t0) 3 ∗(x) = ∗(t ) + x x 0 γ log (96) depends on two parameters has no finite asymptotic value. The acceleration parameter is κ[ ( ) + ( )] 2˙ 2( ) − =− mS t0 W t0 0 ∗ t0 negative q 2, the model is non-physical. H∗ = ,σ= . 3 2[mS(t0) + W(t0)] (88) 4 Effective fermion mass induced by the coupling of the Now we obtain spinor field with axionic dark matter and dynamic ˙ ∗(t0) aether ∗(x) = ∗(t0) + √ 3H∗ σ ⎡ ⎤ 4.1 Reconstruction of the function ∗(S, P,ω,) + x3 + + 1 − ⎢ 1 σ 1 1 σ 1 ⎥ ⎢ ⎥ × log ⎣ ⎦. (89) We discuss here the simplest example of the function ∗(ρ), x3 1 1 + σ − 1 1 + σ + 1 which depends on the argument ρ = τ S2 + τ ω2 + τ (P2 + 2), (97) The corresponding dependence of the scale factor on the cos- 1 2 3 ( ) mological time a t is given by where τ1, τ2, τ3 are some model parameters. For the equi- ρ 1 librium configuration, we obtain from (56), (57), (60) that 2 3 3 √ happens to be proportional to x−6: a(t) = a(t0) H∗(t − t0) + 1 + σ − σ . (90) 2 ρ = ρ( ) −6, t0 x Asymptotic behavior of the scale factor is characterized by 2 2 2 2 ρ(t0) ≡ τ1 S (t0) + τ2ω (t0) + τ3(P (t0) + (t0)) . the power laws (98) 2 3H∗t 3 a(t →∞) → a(t ) , (91) Thus, we can put the quantity 0 2 − 1 ρ(t) 6 and the asymptotic value of the guiding function is x = (99) ρ(t0) + 1 − ˙ ( ) 1 σ 1 into (71), (84), or into (89) obtaining the necessary function ∗ t0 (ρ) τ = τ = τ = ∗(x →∞) = ∗(t0) + √ log . (92) ∗ . Particularly, when 1 1 and 2 3 0, we see 3 σ 1 ρ = 2 ∗ 1 + σ + 1 that S , and the behavior of the guiding function is predetermined by the fermion number density S. When Since the parameter σ is positive (see (88)), the acceleration τ2 = 1 and τ1 = τ3 = 0, the interacting aether and spinor parameter is now monotonic and negative: field regulate the state of the axionic dark matter via the function ∗(ω). x3 + 4σ − q =− . (93) ( 3 + σ) 2 x 4.2 Oscillations of the effective spinor mass, induced by This model can not explain the late-time accelerated expan- the axionic dark matter in the presence of the dynamic sion. aether

3.2.5 Solutions for the model with = 0,W(t0) = 0 and 4.2.1 Scheme of analysis m = 0 We have found the guiding function ∗(ρ) as the exact solu- In this special case the Hubble function (66) tion to the field equations of the model, for which the axionic dark matter is in the first Equilibrium state φ=∗. Since in all κ 2˙ 2( ) −3 0 ∗ t0 the basic formulas (see, e.g., (61)) the spinor mass m appears H(x) = γ x ,γ= , (94) ( ) 6 in the combination with the baryon energy density W t0 , 123 674 Page 10 of 12 Eur. Phys. J. C (2021) 81 :674 as [mS(t0) + W(t0)], in this subsection only, for the sake of By the way, for the massless spinor field, m = 0, and in simplicity, we put W(t0) = 0. Now we consider the effective the absence of a baryonic matter, this quantity behaves as mass matrix of spinor particles based on the formula (22) μ(t) =−μ∗ξ(t), with with ∗=∗(ρ) for the pseudoscalar field φ near the Equi- m2 2˙ ∗(t ) librium (φ = ∗ + ξ). We can estimate the induced mass μ∗ = A 0 0 . ( ) (107) μ ≡M−m as follows: 3S t0 H∞

2 2 T d∗ m d∗ 4.2.4 The case = 0,m= 0 μ =−2 ρ ≈− A 0 x4 ξ(t). (100) S dρ 3S(t0) dx The function H(x) calculated using the formula (89) The function ξ(t) satisfies the linearized equation ˙ 3 ξ¨ + ξ˙ + 2 ξ = . ∗ x 3H m A 0 (101) H(x) =− · √ , (108) H∗ x3 + σ We assume that on the late-time expansion stage H << m A, x →∞ and the third term in (101) seems to be much bigger than the shows that asymptotically, at the spinor mass varia- second one. Thus, if we put the approximate solution tion does not tend to constant, i.e., the model is instable. ˙ ξ(t1) ξ(t) = ξ(t1) cos m A(t − t1) + sin m A(t − t1) (102) m A 5 Discussion and conclusions into (100) we can illustrate the idea about an axionically induced fermion mass oscillations. Clearly, the properties of Whenever we remember the grand event: the detection of H = 4 d∗ neutrinos emitted due to the explosion of Supernova 1987A, the function x dx predetermine the behavior of the effective mass variation μ; below we calculate μ(t) for three we try to imagine what could happen during the 168.000 cases described above. light-years traveling of that neutrinos from the Large Magel- lanic Cloud to the Earth? The neutrinos born in such catas- 4.2.2 Induced mass μ in the case = 0 and β = 1 α2 trophes could interact with dark matter and dark energy, and 4 one can try to find the fingerprints of the cosmic dark fluid In this general case the function H(x) has the form: in the data of neutrino observations. On the other hand, the neutrino flow from the Supernovae of the Type II (SN core- d∗ x3 collapse) is predicted to be so huge, that the neutrinos can H(x) = x4 = , (103) dx x6 + αx3 + β influence the state of dark matter in the source environment. Keeping in mind this idea, we established the model of thus, taking into account (73) we obtain interaction between spinor field, axionic dark matter and 2 2 ˙ m (t0) dynamic aether. The central element of this model is the so- μ(t) =−ξ(t) A 0 ∗ 3S(t0) called guiding function , which is the detail of the potential ⎧ α ⎫ ⎨sinh [3H∞(t − t )] − √ ⎬ of the pseudoscalar (axion) field (see (8)). Since this guid- 0 β−α2 × 4 . ing function depends on the spinor field and dynamic aether ⎩ ⎭ (104) cosh [3H∞(t − t0)] via some scalars (we considered here four scalar (9)), we are faced with the nonlinear modifications of the equations for H( ) → In the asymptotic regime x 1. If the spinor field is the vector, axion, spinor and gravitational fields. We have = α = massless, m 0, we see that 0 and we deal with the found the exact solutions to these modified field equations formula in the model with equilibrium axionic dark matter, in the m2 2(˙ t ) framework of the isotropic homogeneous cosmology. For the μ( ) =−ξ( ) A 0 0 ( − ) . t t tanh [3H∞ t t0 ] (105) model with nonvanishing cosmological constant the exact 3S(t0) solutions for the guiding function are presented by (71) and = 4.2.3 Induced mass μ in the case = 0 and β = 1 α2 (84); when 0, the corresponding exact solutions are of 4 the form (89) and (96). In this special case, according to (84) and (83), the spinor The influence of the dynamic aether reveals in the mod- mass variation is given by the formula ification of the model working parameters by the factor , which contains Jacobson’s phenomenological constants (62). 2 2 ˙ m ∗(t0) The constraint on the sum of two parameters C + C has μ =−ξ(t) A 0 1 3 ( ) 3S t0 H∞ been obtained in 2017 due to the observation of the binary κmS(t0) − ( − ) neutron star merger (the event encoded as GW170817 and × 1 − e 3H∞ t t0 . (106) κmS(t0) + GRB 170817A, see [61]). It was established that the ratio of 123 Eur. Phys. J. C (2021) 81 :674 Page 11 of 12 674 the velocities of the gravitational and electromagnetic waves the dark matter influence. In contrast to this reference infor- differs from one by the quantity about 10−15, and thus, the mation, one can try to measure the corresponding time delay sum of the parameters C1 + C3 is estimated as follows: in the Supernova burst search; now the supernova neutrinos −15 −15 −6 × 10 < C1 + C3 < 1.4 × 10 . If we assume that are assumed to be influenced by the axionic dark matter in − = + 3 C3 C1,wehavetowrite 1 2 C2 and to redefine the long trip to the Earth. Comparison of the corresponding correspondingly the parameter H∞. estimation of the neutrino mass taking into account the axion- Then we use the obtained exact solutions for ∗ for find- ically induced oscillations, with the reference one, could help ing of the so-called effective spinor mass (see (21)forthe to test our hypothesis. effective spinor mass matrix, and (22) for the effective mass scalar). The results of calculations, which demonstrate the Acknowledgements This work was supported by the Russian Foun- possibility of the induced spinor mass oscillations in the dation for Basic Research (Grant no. 20-52-05009). cosmological context, are presented by the formulas (104)– Data Availability Statement This manuscript has no associated data or (106). Clearly, the parameter μ∗ defined by (107) plays the the data will not be deposited. [Authors’ comment: The paper presents role of effectiveness of oscillation production. We have to the results of completely theoretical research; it has no associated specific data; all the necessary experimental information can be found in emphasize that for finding of the guiding function ∗ and of the references [61]and[62].] the induced spinor mass μ we do not calculate directly the components of the spinor field, but we have found the exact Open Access This article is licensed under a Creative Commons Attri- solutions for the spinor scalars (pseudoscalars) S, P, ω and bution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you , which are the arguments of the guiding function ∗ (see give appropriate credit to the original author(s) and the source, pro- the details in Sect. 3.1.5). vide a link to the Creative Commons licence, and indicate if changes The obtained results allow us to formulate the following were made. The images or other third party material in this article three main conclusions. are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended 1. The cosmic spinor field influences the state of the axionic use is not permitted by statutory regulation or exceeds the permit- dark matter via the guiding function ∗ (the argument of ted use, you will need to obtain permission directly from the copy- the periodic potential of the axion field V (φ, ∗)), which right holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. is a time depending analog of the vacuum average value Funded by SCOAP3. of the pseudoscalar field. 2. Spinor particles (massive and massless) acquire effective masses due to the interaction with the axionic dark matter; variations of this effective mass are predetermined References by the dynamics of the Universe expansion. 3. The effective mass of the spinor field is proportional to 1. R.N. Mohapatra, P.B. Pal, Massive Neutrinos in Physics and Astro- the square of the axion mass, is inversely proportional physics, Lecture Notes in Physics (World Scientific, Singapore, to the square of the constant of the axion-photon cou- 1991) 2. C. Giunti, C.W. Kim, Fundamentals of Neutrino Physics and Astro- pling, depends on cosmological time, and is exposed to physics (Oxford University Press, Oxford, 2007) oscillations with the frequency proportional to the axion 3. D.J.E. Marsh, Axion cosmology. Phys. Rep. 643, 1–79 (2016) mass m A. 4. M. Khlopov, Fundamentals of Cosmic Particle Physics (CISP- Springer, Cambridge, 2012) 5. A. Del Popolo, Non-baryonic dark matter in cosmology. Int. J. The final question, which we would like to touch, concerns Mod. Phys. D 23, 1430005 (2014) the possibility to observe the predicted effect of axionically 6. F. Chadha-Day, J. Ellis, D.J.E. Marsh, Axion dark matter: what is induced spinor mass oscillations. According to the recent it and why now? arXiv:2105.01406 report [62], the unique new Hyper-Kamiokande detector is 7. T. Jacobson, D. Mattingly, Gravity with a dynamical preferred frame. Phys. Rev. D 64, 024028 (2001) announced to be able to monitor the atmospheric, supernova 8. T. Jacobson, Einstein-aether gravity: a status report. PoSQG-Ph and solar neutrinos. We do not discuss the details and design 020, 020 (2007) of a new test, but its idea could be the following. The distance 9. C. Heinicke, P.Baekler, F.W.Hehl, Einstein-aether theory, violation between the Sun and Earth is much less than the distance of Lorentz invariance, and metric-affine gravity. Phys. Rev. D 72, 025012 (2005) between the Supernova and Earth. If the detecting system 10. V. Sahni, A. Starobinsky, Reconstructing dark energy. Int. J. Mod. is able to measure the time difference between a solar flare Phys. D 15, 2105–2132 (2006) and a solar neutrino appearance in the terrestrial laboratory, 11. J. Frieman, M. Turner, D. Huterer, Dark energy and the accelerating it could give the difference between the velocities of the pho- universe. Ann. Rev. Astron. Astrophys. 46, 385–432 (2008) 12. K. Bamba, S. Capozziello, S.D. Odintsov, Dark energy cosmology: tons and neutrinos, and thus could be considered as a standard the equivalent description via different theoretical models and cos- for the estimation of the neutrino mass in the model free of mography tests. Astrophys. Space Sci. 342, 155–228 (2012) 123 674 Page 12 of 12 Eur. Phys. J. C (2021) 81 :674

13. C.D.R. Carvajal, B.L. Sánchez-Vega, O. Zapata, Linking axionlike 40. B. Saha, V. Rikhvitsky, Anisotropic cosmological models with dark matter to neutrino masses. Phys. Rev. D 96, 115035 (2017) spinor and scalar fields and viscous fluid in presence of a term: 14. T.B. Watson, Z.E. Musielak, Chiral symmetry in Dirac equation qualitative solutions. J. Math. Phys. 49, 112502 (2008) and its effects on neutrino masses and dark matter. Int. J. Mod. 41. B. Saha, Interacting scalar and spinor fields in Bianchi type I uni- Phys. 35, 2050189 (2020) verse filled with magneto-fluid. Astrophys. Space Sci. 299, 149– 15. L.M. Krauss, S. Nasri, M. Trodden, A model for neutrino masses 158 (2005) and dark matter. Phys. Rev. D 67, 085002 (2003) 42. B. Saha, G.N. Shikin, Interacting spinor and scalar fields in Bianchi 16. E. Ma, Verifiableradiative seesaw mechanism of neutrino mass and type-I universe filled with perfect fluid: exact self-consistent solu- dark matter. Phys. Rev. D 73, 077301 (2006) tions. Gen. Relativ. Gravit. 29, 1099–1113 (1997) 17. C. Boehm, Y. Farzan, T. Hambye, S. Palomares-Ruiz, S. Pascoli, 43. B. Saha, Nonlinear spinor fields in Bianchi type-I spacetime: prob- Is it possible to explain neutrino masses with scalar dark matter? lems and possibilities. Astrophys. Space Sci. 357, 28 (2015) Phys. Rev. D 77, 043516 (2008) 44. B. Saha, Non-minimally coupled nonlinear spinor field in Bianchi 18. Y. Farzan, A Minimal model linking two great mysteries: neutrino type-I cosmology. Eur. Phys. J. Plus 134, 491 (2019) mass and dark matter. Phys. Rev. D 80, 073009 (2009) 45. S.D. Odintsov, V.K. Oikonomou, Unification of inflation with dark 19. T. Asaka, S. Blanchet, M. Shaposhnikov, The nuMSM, dark matter energy in f(R) gravity and axion dark matter. Phys. Rev. D 99, and neutrino masses. Phys. Lett. B 631, 151 (2005) 104070 (2019) 20. A. Berlin, D. Hooper, Axion-assisted production of sterile neutrino 46. S.D. Odintsov, V.K. Oikonomou, Neutron stars phenomenology dark matter. Phys. Rev. D 95, 075017 (2017) with scalar-tensor inflationary attractors. Phys. Dark Universe 32, 21. A.R. Khalifeh, R. Jimenez, Spinors and scalars in curved space- 100805 (2021) time: neutrino dark energy (DEν). Phys. Dark Universe 31, 100777 47. S. Nojiri, S.D. Odintsov, Unified cosmic history in modified grav- (2021) ity: from F(R) theory to Lorentz non-invariant models. Phys. Rep. 22. A. Capolupo, S. Capozziello, G. Vitiello, Neutrino mixing as a 505, 59 (2011) source of dark energy. Phys. Lett. A 363, 53–56 (2007) 48. S. Nojiri, S.D. Odintsov, V.K. Oikonomou, Modified gravity theo- 23. A.W. Brookfield, C. van de Bruck, D.F. Mota, D. Tocchini- ries on a nutshell: inflation, bounce and late-time evolution. Phys. Valentini, Cosmology with massive neutrinos coupled to dark Rep. 692, 1 (2017) energy. Phys. Rev. Lett. 96, 061301 (2006) 49. S. Weinberg, Effective field theory for inflation. Phys. Rev. D 77, 24. R.D. Peccei, Neutrino models of dark energy. Phys. Rev. D 71, 123541 (2008) 023527 (2005) 50. M. Gomes, J.R. Nascimento, AYu. Petrov, A.J. da Silva, On the 25. M.C. Gonzalez, Q. Liang, J. Sakstein, M. Trodden, Neutrino- aether-like Lorentz-breaking actions. Phys. Rev. D 81, 045018 assisted early dark energy: theory and cosmology. Phys. Rev. D (2010) 94, 083519 (2016) 51. A.J.G. Carvalho, D.R. Granado, J.R. Nascimento, AYu. Petrov, 26. A. Addazi, S. Capozziello, S. Odintsov, Born–Infeld condensate as Non-abelian aether-like term in four dimensions. Eur. Phys. J. C a possible origin of neutrino masses and dark energy. Phys. Lett. B 79, 817 (2019) 760, 611–616 (2016) 52. A.B. Balakin, G.B. Kiselev, Spontaneous color polarization as a 27. A. Basak, J.R. Bhatt, Lorentz invariant dark-spinor and inflation. modus originis of the dynamic aether. Universe 6(7), 95 (2020) JCAP 06, 011 (2011) 53. R.D. Peccei, H.R. Quinn, CP conservation in the presence of instan- 28. C.G. Boehmer, J. Burnett, D.F. Mota, D.J. Shaw, Dark spinor mod- tons. Phys. Rev. Lett. 38, 1440–1443 (1977) els in gravitation and cosmology. JHEP 2010(7), 53 (2010). https:// 54. S. Weinberg, A new light boson? Phys. Rev. Lett. 40, 223–226 doi.org/10.1007/JHEP07(2010)053 (1978) 29. J. Lee, T.H. Lee, Ph Oh, Conformally-coupled dark spinor and 55. F. Wilczek, Problem of strong P and T invariance in the presence FRW universe. Phys. Rev. D 86, 107301 (2012) of instantons. Phys. Rev. Lett. 40, 279–282 (1978) 30. A.R. Solomon, J.D. Barrow, Inflationary instabilities of Einstein- 56. V. Fock, D. Iwanenko, Geometrie quantique lineaire et deplace- aether cosmology. Phys. Rev. D 89, 024001 (2014) ment parallele. Comptes Rendus Acad Sci. (Paris) 188, 1470–1472 31. J.D. Barrow, Some inflationary Einstein-aether cosmologies. Phys. (1929) Rev. D 85, 047503 (2012) 57. A.B. Balakin, D.E. Groshev, Polarization and stratification of 32. A.B. Balakin, A.F. Shakirzyanov, Axionic extension of the axionically active plasma in a dyon magnetosphere. Phys. Rev. Einstein-aether theory: how does dynamic aether regulate the state D 99(2), 023006023006 (2019) of axionic dark matter? Phys. Dark Universe 24, 100283 (2019) 58. A.B. Balakin, D.E. Groshev, New application of the Killing vector 33. A.B. Balakin, A.F. Shakirzyanov, Is the axionic dark matter an field formalism: modified periodic potential and two-level profiles equilibrium system? Universe 6(11), 192 (2020) of the axionic dark matter distribution. Eur. Phys. J. C 80(2), 145 34. A.B. Balakin, Axionic extension of the Einstein-aether theory. (2020) Phys. Rev. D 94(2), 024021 (2016) 59. A.B. Balakin, Extended Einstein–Maxwell model. Gravit. Cosmol. 35. A.B. Balakin, J.P.S. Lemos, Einstein-aether theory with a Maxwell 13(3(51)), 163–177 (2007) field: general formalism. Ann. Phys. 350(11), 454–484 (2014) 60. A.B. Balakin, Magnetic relaxation in the Bianchi-I universe. Class. 36. TYu. Alpin, A.B. Balakin, Birefringence induced by pp-wave Quantum Gravity 24(20–21), 5221–5245 (2007) modes in an electromagnetically active dynamic aether. Eur. Phys. 61. LIGO Scientific Collaboration, Virgo Collaboration, Fermi J. C 77(10), 699 (2017) Gamma-Ray Burst Monitor, INTEGRAL, gravitational waves and 37. S. Bertolini, L. Di Luzio, H. Kolesová, M. Malinský, J.C. gamma-rays from a binary neutron star merger: GW170817 and Vasque, Neutrino-axion-dilaton interconnection. Phys. Rev. D 93, GRB 170817A. APJ Lett. 848, L13 (2017) 015009015009 (2016) 62. K. Abe et al. (Hyper-Kamiokande Proto-Collaboration), Hyper- 38. A.B. Balakin, The extended Einstein–Maxwell-aether-axion Kamiokande design report. arXiv:1805.04163 model: exact solutions for axionically controlled pp-wave aether modes. Mod. Phys. Lett. A 33(9), 1850050 (2018) 39. T.Y.Alpin, A.B. Balakin, The Einstein–Maxwell-aether-axion theory: dynamo-optical anomaly in the electromagnetic response. Int. J. Mod. Phys. D 25(4), 1650048 (2016) 123