3.1. Bosonic Case
Let us first consider the case when a baryonic number is carried by a complex scalar field
[
17]. The total action has the form:
where
is the potential of field
and
is the matter action which does not include the field
. In Equation (
5),
is the classical curvature field, while
is the quantum operator of light scalar particles.
We assume that the potential
is not invariant with respect to phase transformation
and thus the corresponding current
is not conserved. Here,
q is the baryonic number of field
. The nonconservation of the current is necessary for the proper performance of the model, otherwise
in Equation (
3) can be integrated away by parts.
Varying action (
5) over
, we come to the following equations:
where
is the covariant derivative in metric
(of course, for scalars
) and
is the energy–momentum tensor of matter obtained from action
.
Taking the trace of Equation (
7) with respect to
and
and changing the sign, we obtain:
where
is the trace of the energy–momentum tensor of matter. For the usual relativistic matter,
, while for scalar field
, the trace of the energy–momentum tensor is nonzero:
The equation of motion for field
is:
According to definition (
6) and Equation (
10), the current divergence is:
For a homogeneous curvature scalar
in a spatially flat FLRW-metric
Equation (
8) is reduced to:
where
is the baryonic number density of the
-field,
is the Hubble parameter, and the divergence of the current is given by the expression:
As we see in what follows, the last two terms in Equation (
13) do not have an essential impact on the cosmological instability found in Ref. [
17] and are disregarded below. Indeed, as shown in Ref. [
17], the field
does not exponentially rise together with
R and thus can be neglected in comparison with the terms containing
R. In the case considered here of a modified
-gravity, the curvature also initially strongly rises before the
term starts to operate and, in this sense, the situation is the same as that studied in Ref. [
18]. In fact, Equation (
15) is a good argument in favour of the subdominant nature of the terms containing
above.
Let us note that the statement of exponential instability of
[
17] does not depend on the conservation or nonconservation of the current from the potential term
in Equation (
14). However, if the current from this term is conserved, then the baryon asymmetry is not generated. On the other hand, the term in square brackets in Equation (
14) does not lead to the generation of the baryon asymmetry but leads to the exponential instability of
. Below, we ignore the last term of Equation (
14).
Performing thermal averaging of the normal-ordered bilinear products of field
in the high temperature limit (see Appendix of Ref. [
17]) in accordance with equations:
and using Equation (
14), we obtain the fourth-order differential equation:
Here, is the thermal average value of the baryonic number density of , which is supposed to vanish initially, but created through the process of gravitational baryogenesis. This term can be neglected because the baryon asymmetry is normally quite small. Even if it is not small, it does not have a considerable impact on the explosive rise of the curvature scalar. As we see in what follows, the evolution of proceeds much faster than the cosmological evolution, that is . Consequently, we neglect the terms proportional to R with respect to the terms proportional to the second derivative of R, . We also consider the terms of the type as small with respect to . We can check that this presumption is true a posteriori with the obtained solution for .
Keeping only the dominant terms we simplify the above equation to:
where
While studying the instability of the solution, we do not take into account the r.h.s. of Equation (
17) which does not depend upon R. Looking for the solution of Equation (
17) in the form
, we obtain the characteristic equation:
with the eigenvalues
defined by the expression:
There is no instability if
and Equation (
17) has only oscillating solutions. It is realised if
. Using the expression in Equation (
18) for
and taking
GeV, we find the stability condition
which is fulfilled for all interesting values of
M.
The value of depends upon the relation between and . If , then the frequency of the oscillations of curvature is of the order of and . If , then there are two possible solutions and . High-frequency oscillations of R would lead to an efficient gravitational particle production and, as a result, to a damping of the oscillations.
In fact, both conditions and are essentially the same at the stage of exponential rise of , since the r.h.s in both cases is just . Since H drops down with decreasing temperature and on the opposite rises up, these conditions should be true at sufficiently small temperatures.
3.2. Fermionic Case
In this section, we consider the case when a baryonic number is carried by fermions. The gravitational part of the action has the form as in Equation (
4), while the fermionic part of the action is the same as in Refs. [
10,
18]:
where
Q is the quarklike field with nonzero baryonic number
,
is the charged conjugated quark operator,
L is another fermionic field (lepton), and
is the covariant derivative of the Dirac fermions in tetrad formalism. The quark current is
with
being the curved space gamma matrices, and
describes all other forms of matter. The four-fermion interaction between quarks and leptons is introduced to ensure the necessary nonconservation of the baryon number with
being a constant parameter with dimension of mass and
g being a dimensionless coupling constant. In the term describing the interaction of the baryonic current of fermions with the derivative of the curvature scalar,
M is a constant parameter with a dimension of mass and
is a dimensionless coupling constant which is introduced to allow for an arbitrary sign of the above expression.
Gravitational equations of motion with an account of
-term in analogy with Equation (
7) take the form:
Taking the trace of Equation (
23) with an account of fermion equations of motion, we obtain:
where
is the trace of the energy momentum tensor of all other fields. In the early universe when various species are relativistic, we can take
. The average expectation value of the quark–lepton interaction term proportional to
g is also small, so the contribution of all matter fields may be neglected and hence the only term which remains in the r.h.s. of Equation (
24) is that proportional to
.
A higher-order differential equation for
R is obtained after we substitute the current divergence,
, calculated from the kinetic equation in the external field
R [
18], into Equation (
24). For the spatially homogeneous case,
where the collision integral,
, in the lowest order of perturbation theory is equal to:
Here,
is the amplitude of the transition from state
a to state
b,
is the baryonic number of quark,
is the phase-space distribution (the occupation number), and
where
is the energy of a particle with three-momentum
q and mass
m. The element of the phase space of the final particles,
, is defined analogously.
We choose such representation of the quark operator,
Q, for which the interaction of the baryonic current with the derivative of the curvature scalar in Equation (
22) vanishes but reappears in the quark–lepton interaction term:
We make the simplifying assumption that the evolution of
R can be approximately described by the law
We assume that
slowly changes at the characteristic time scale of the reactions, which contribute to the collision integral (
26), and so we can approximately take
.
According to the rules of quantum field theory, the reaction probability is given by the square of the integral over space and time of the amplitude of the corresponding process. In the case of a time-independent interaction, it leads to the energy conservation,
. If the interaction depends upon time, the energy evidently is nonconserved and in our case, e.g., for the reaction
, the energy balance has the form:
In kinetic equilibrium, the phase-space distribution of fermions has the form
where
is the dimensionless chemical potential, different for quarks,
, and leptons,
. In the thermal equilibrium case, the condition of conservation of chemical potentials is fulfilled, that is,
. In particular, it demands that chemical potentials of particles and antiparticles are equal by magnitude and have opposite signs:
, as follows, e.g., from the consideration of particle–antiparticle annihilation into different numbers of photons. If energy is not conserved, due to the time-dependent
, the conservation of chemical potentials is also broken, as we see in what follows.
We assume that
and hence, distribution (
31) turns into:
We also assume that
and correspondingly, the balance of chemical potentials in equilibrium for the reactions
leads to:
Following Ref. [
18], we express
where
is the number of quark spin states. Since we are studying the instability of
R whose timescale is presumed to be much smaller than the expansion rate of the Universe, we approximate
is obtained from Equation (
33), using the conservation of the sum of baryonic and leptonic numbers, which implies
. Then,
Substituting Equation (
37) in Equation (
36) and neglecting the
-term, Equation (
24) gives the following fourth-order differential equation for the curvature scalar:
where
Once again, we consider terms containing
R as small with respect to the terms containing
. The value of
is only slightly numerically different from
in Equation (
18) and has the same dependence upon the essential parameters, so the solutions of Equations (
17) and (
38) practically coincide.